Application of probabilistic techniques in flood protection in the Netherlands

Application of probabilistic techniques in flood

protection in the Netherlands

Table of contents

1. Introduction ...3

2. Safety standards ...4

3. Organisation of flood protection...5

3.1 Legislation...5

4. Design of coastal dikes ...6

4.1 Introduction...6

4.2 Structural design procedure ...6

4.2.1 Traditional procedure ...6 4.2.2 Sophisticated procedure...7 4.3 Probabilistic procedure...8 4.4 Example ... 10 5. Design of dunes ... 13 5.1 Introduction... 13

5.2 Normative probability of failure ... 14

5.3 Safety assessment of a cross section of a dune coast ... 14

5.3.1 Introduction ... 14

5.3.2 The simple safety assessment method... 15

5.3.3 Computational model / erosion profile ... 21

5.4 Example ... 23

6. Developments ... 26

6.1 Probabilism ... 26

6.1.1 Example ... 26

6.2 Safety philosophy based on flooding risks ... 28

1. Introduction

About a quarter of the Netherlands is below mean sea le vel. Without flood protection structures, about two-third of the country would be flooded during storm surges at sea or high discharges in the rivers (see Error!

Unknown switch argument.). That is why in a country such as the Netherlands, protection against flooding is

an important task. This protection is provided by an extensive system of so -called primary flood protection structures.

In this study attention is especially given to flood protection due to storm surges at sea.

Figure Error! Unknown switch argument. The Netherlands without flood protection structures

The area which is protected by a linked system of primary flood protection structures is called a dike ring area (dijkringgebied). The flood protection structures around a dike ring area can be divided into sections, in which load and strength characteristics are comparable. These sections can consist of dikes, dunes, structures or high grounds. High grounds are areas which are high enough and thus don’t need protection against flooding. Together these sections ensure the safety of the area.

The present safety standards for the flood protection structures are expressed as return periods of extreme water levels, which the flood protection structure must be able to withstand (see paragraph 2). Hydraulic boundary conditions are associated to these safety standards. These boundary conditions together with technical guidelines are used in designing coastal dikes an d dunes (see paragraph 4 and 5). The situation described is largely based on the standard of practice in the Netherlands. The state of the art however shows a number of developments which may be introduced in the standard of practice in the near future. These

developments are described In paragraph 6.

2. Safety standards

In determining the required height of dikes, the traditional method in the Netherlands used until well into this century was to take the highest known water level, plus a margin of 0.5 to 1 metre. The Delta Commission, which was set up shortly after the disastrous floods of 1953, laid down the basis in 1956 for the current safety standards with regard to protection against flooding. The starting point as proposed by the Delta Commission was to establish a desired level of safety for each dike ring area or polder. This safety level would need to be based on the costs of construction of dikes and on the possi ble damages which would be caused by flooding. These economic analysis led to an ‘optimum’ safety level expressed as the probability of failure for the coastal dikes. In practice however, the safety level was expressed as the return period of the water level, being the most dominant hydraulic load. One of the main reasons to simplify the description of the safety standard was the lack of knowledge to describe the failure process of a dike sufficiently accurate.

This economic analysis led to an ‘optimum’ safety level expressed as the probability of failure for the coastal dikes. In practice however, the safety level was expressed as the return period of the water level, being the most dominant hydraulic load. One of the main reasons to simplify the description of the safety standard was the lack of knowledge to describe the failure process of a dike sufficiently accurate.

The economic analysis has been used to differentiate the safety standard according to the expected damages in the various polders. A safety standard has been established for each dike ring area. This standard is expressed as the mean yearly frequency that the prescribed flood level is being exceeded. The standards vary from 1/10000 to 1/1250 per year (see Error! Unknown switch argument.), depending on the economic activities and size of population in the protected area, and the nature of the threat (river or sea ). In 1996 these standards were laid down in legislation when the Flood protection Act (Ministry of Transport, Public works and Water Management 1996) came into effect. The flood levels associated to the safety standards are updated every five years to accommodate sea level rise and recent technical developments.

In designing the dike, a certain margin with regard to the flood level is applied, depending on wind and wave conditions. The object of this margin is to ensure that each individual dike section i s sufficiently high to withstand the prescribed flood levels and associated hydraulic loads. Technical guidelines give the engineer sufficient information to calculate the required margins and other structural aspects of the dike design. Figure Error! Unknown switch argument. Safety standards in the

3. Organisation of flood protection

The care of the flood protection structures in the Netherlands is divided over three forms of government: the State, the provinces and the waterboards. The municipalities play a part in the town and country planning (as a representative for the other interests concerning flood protection structures such as living and traffic) and in the case of a threatening calamity.

A central role has been assigned to the waterboards. A waterboard is a so -called functional form of

government, aimed on watermanagement and the management of flood protection structures. The province (the Netherlands has 12 provinces) has a task of supervision over municipalities as well as waterboards.

3.1 Legislation

The goal of the Flood Protection Act is to legally anchor the safety against flooding by water from the sea or from the rivers. The Flood Protection Act gives safety standards for all the dike ring areas. This has been legally anchored, because the longer ago a flooding has occurred, the less the societal awareness of the risks of flooding. The Flood Protection Act obliges the manager of a flood protection structure to check the flood protection structure every 5 years and to report this in relation to the safety standard of that particular flood protection structure.

4.

Design of coastal dikes

4.1 Introduction

The structural design of a coastal dikes can be characterised mainly by determining the required crest level, stability of the revetments, geotechnical stability and the reliability of movable objects intersecting the dike, such as sluices. Although the crest level is not the single required safety feature of the dike, the scope of the study will be limited to this aspect. In a follow-up other safety features can be studied.

The crest level of the dikes needs to be sufficient to prevent too large quantities of water overtopping the structure. Two conditions may be of importance :

 overtopping without failure of the dike, leading to too large quantities of water in the polder;

 overtopping leading to failure of the dike due to erosion or geotechnical instability (infiltration) of the inner slope.

Whether the first condition is of importance depends very much on the local situation. The study will focus on the second condition which has a general character. Two desi gn procedures for determining the required crest level are available : a traditional method which has been applied to majority of the coastal dikes and a more sophisticated method which has been developed quite recently. Both methods are described in paragraph 4.2. In paragraph 4.3 a probabilistic procedure to find the hydraulic boundary conditions for the design is shown. An example of the design of a coastal dike is elaborated in paragraph 4.4.

4.2 Structural design procedure

4.2.1 Traditional procedure

The required crest level is calculated using the following procedure :  crest level = design water level + wave run-up + additional margins

 the design water level is the flood level with the legally prescribed return period; the flood level is updated every five years

(1) z2% = 2% wave run-up (m)

Hs = significant wave height (m)

tan = steepness of the outer slope (-)

 additional margins are to compens ate for sea level rise, settlement and seiches sea level rise = 20 cm per century

settlement = based on geotechnical calculation

seiches = depending on local situation, ranging from 10 cm to 80 cm

The significant wave height is traditionally determined by calculating the expected significant wave height during the design flood. For the calculation of wind -generated waves the Delta Commission issued design wind velocities ranging from 30 to 35 m/s. Swell is related to the flood level. The peak period o r the shape of the wave energy spectrum is not taken into account.

The design lifetime of a coastal dike is generally 50 years, so the margin for sea level rise is in most cases 10 centimetres. The required margin for settlement is based on a prediction of the settlement during 50 years.  wave run-up : z2% = 8Hstan

4.2.2 Sophisticated procedure

4.2.2.1 Sophisticated procedure with regard to wave run-up

The sophisticated procedure contains a more sophisticated wave run -up formula. The other items of the procedure remain unchanged.

(2) Where:







= surf similarity parameter (-)

Tp = wave period (s)

 is a reduction factor which takes the effects of friction, foreshores, angular wave attack and the presence of a berm into account: =f h  b.

The reduction factor f varies between 0.5 for rock slopes with two or more layers, 0.6 for rock slopes with one

layer, 0.95 for grass and 1.0 for smooth impermeable slopes.

The reduction factor h can be approximated by h = H2%/(1.4 Hs). Van der Meer (1997) proposed not to use the

reduction factor for foreshores.

The reduction factor  can be approximated by  = 1-0.0022. (  800).

Due to the presence of a berm in the seaward slope the slope angle in the surf-similarity parameters is not uniquely defined. The influence of a berm can be taken into account by replacing tan in the surf-similarity parameter by tan = rdhtan+(1-rdh)taneq where the weight factor rdh depends on the position of the berm:

rdh=0.5(dh/Hs) 2

where dh is the absolute value of the average depth of the berm with respect to the still -water

level (-1.0  dh/Hs  1.0); tan describes the slope above and below the berm and taneq describes an

equivalent slope angle around the berm defined as: taneq=2/(2cot+B/Hs) where B is the berm width. Use is

made of the reduction factor b=tan/tan (b0.6).

Since limited information is available on combinations of reductions, a maximum total reduction factor =f h  b = 0,4 was proposed.

The coefficients c0, c1 and p were set at 1.5, 3 and 2 based on many physical -model tests. For design purposes

it was advised to use somewhat safer values: 1.6, 3.2 and 2 respectively.

Data on which the above-mentioned approach was based is shown in Error! Unknown switch argument..  wave run-up : z2% / (Hs) = c0op for op p

Figure Error! Unknown switch argument. Measured wave run-up levels and formula proposed by De Waal and Van der Meer (1992)

4.2.2.2 Sophisticated procedure with regard to wave overtopping

The sophisticated procedure can be extended by introducing wave overtopping instead of wave run -up. Again, the other items of the procedure remain more or less unchanged. The required crest level is now calculated using the following procedure :

 crest level = design water level + margin for wave overtopping + additional margins

(3)

The maximum overtopping rate is :Q c c R H MAX c d c s        exp



= non-dimensional (critical) overtopping discharge

q = overtopping discharge (m3/s/m)

Rc = crest elevation with respect to the design water level (m)

The reduction factor  takes the effects of friction, foreshores, angular wave attack and the presence of a berm into account: =f h b.

The reduction factors can be taken the same as for wave run -up except for the influence of the angle of wave attack for which a slightly different formula was proposed:  = 1-0.0033. (  800).

As a best-fit through the data-points the coefficients ca, cb, cc and cd were set at 0.06, 5.2, 0.2 and 2.6. For

design purposes it was advised to use somewhat safer values: 0.06, 4.7, 0.2 and 2.3 respectively. The design criteria used for overtopping depend on the quality of the inner slope. Critical discharges are :  0.1 l/m/s, with no specific demands to the inner slope with regard to erosion or infiltration

 1 l/m/s/, which requires good quality clay and grass cover with a slope not steeper than 1:2

 10 l/m/s, which requires excellent good quality clay and grass cover with a slope not steeper than 1:3. For coastal dikes the application of 10 l/m/s is being considered at present. This criterion is used for river dikes where wave periods are typically 3 to 4 seconds. For coastal dikes with wave periods up to 10 seconds and more this criterion may lead to enormous volumes of overtopping during shorter periods. Therefore it is considered safety to limit overtopping rates to 1 l/m/s. The traditional 2% wave run -up criterion leads to overtopping rates of 3 to 5 l/m/s.

4.3 Probabilistic procedure

The boundary conditions or hydraulic loads taken into account are more and more resulting from probabilistic analysis. The Delta Commission performed the first probabilistic analysis by introducing the flood level frequency curves and safety standards in terms of return periods. Following the work of the Delta Commission the (joint) statistics of flood level, wind and waves (height and period) are introduced slowly into practice. Theoretically the probability of failure of a coastal dikes can be calculated using the probability density functions of both the loads and strength of the dike and a limit state function which describes the failure in terms of load and strength.  overtopping :

Q

c

c

R

H













_



cot

 

exp

 

Let the strength be : R (in French : résistance) Let the load be : S (in French : sollicitation)

The limit state function can be defined as : Z = R-S. The structure is considered to be in a limit state (on the edge of failing) if Z equals to 0 :

 Z > 0 safe area  Z = 0 limit state  Z < 0 unsafe area.

The load and strength of a coastal dike can be expressed in so called basic variables. These variables can be stochastic or deterministic. Using the traditional design procedure the limit state function can be expressed as :





The crest level (hcrest) is either a running variable (while designing) or a deterministic parameter (while

assessing the actual safety). In both cases this parameter can be treated as a deterministic parameter. The load is the combination of flood level (h) and significant wave height (Hs). The parameters h en Hs are

stochastic variables in this case. The probability density functions of these variables can be described by :  f(h)

 f(Hs).

If the variables are not correlated, the joint probability density function can be calculated quite simple :  f(h)*f(Hs).

This joint probability density function can be shown in a graph using iso -density charts in the h-Hs space (s ee Error! Unknown switch argument.).

The probability of failure is the contents of the joint probability density function in the unsafe area.

P

f h f H dhdH





( ) (

)

The maximum probability P is prescribed in the Flood Protection Act. The combination of h -Hs with the

maximum probability density is called the ‘design point’. In practice this design point will be used by engineers or authorities to design a dikes or to assess the safety.

In general the probabilistic procedure can be described for more complex situations as :

P

f h f h

f

dhdH

d







...

( ) ( )... (.. )

.... ..

The number of basis variables can be extended according to the specific situation and design procedure. Correlation between basic variables can be introduced. But basically the probabilistic procedure remains the same. This means that for every situation simple design points (combinations of basic variables) can be given. In the collected Hydraulic Boundary Conditions for Primary Flood Protection Structures issued by the ministry these design points are given. Only for specific situations (e.g. cost optimisation or tailor made structures) the probabilistic procedure is carried out in the design process.

The probabilistic procedure described before has become more or less general practice in designing flood protection structures. However, probabilism is still largely confined to the hydraulic loads. The strength of the structure and design criteria are mostly taken into account in a deterministic way, using a safety factor which is largely based on practice and engineering judgement. Also the uncertainty of loads, strength, criteria, models and so on is not taken into account as well.

4.4 Example

As an example for the calculation of the required crest level of a coastal dikes the Pettemer Zeewering has been selected. This dike is situated on the Dutch shore as shown in figure 5.

For this dike ring area or polder the safety standard is 1/10.000 per year (see Error! Unknown switch

argument.). The prescribed hydraulic boundary conditions for this location are :





.

These boundary conditions are only partially probabilistic derived. The water level has a return period of 10.000 years. The wave height is the expected wave height associated to this water level. It is not a design

combination derived using joint statistics. The wave height is the wave at the toe of the dike.

Traditional procedure

The traditional design procedure is applied using the straightforward formula : Figure Error! Unknown switch argument. Location Pettemer Zeewering

 hcrest = h+8Hstan

The Pettemer Zeewering has a berm in the outer slope with different slope angles above and below this berm. For this situation an equivalent slope is determined using the following method:

The cross section of the Pettemer Zeewering is shown in Error! Unknown switch argument.. Above the berm the outward facing slope is 1:3,19 and below the berm the slope is 1:4,12. The berm is approximately at storm surge level. This leads to an equivalent slope of 1:4,95.

Figure Error! Unknown switch argument. Cross section Pettemer Zeewering

Using this slope and the prescribed boundary conditions the required crest level is :  hcrest = h+8Hstan

 hcrest = 4,70 m + 8*4,70 m*(0,202) = 12,3 m.

The additional margins for sea level rise, land subsidence (0,1 m) and seiches (0,15 m) lead to a required crest level of MSL+12.55 meter. The present crest level is MSL+12.75 meter, which seems to be sufficient for the moment.

The more sophisticated procedure using wave run-up formula (2) yields quite different results. The effect of the peak wave period of the waves is significant. The peak period used in this example is 12 seconds.

Using formula (2) the required crest level is :  hcrest = h+c0 op  Hs

 hcrest = 4,70 m + 1,6*1,39*0,82*4,70 m = 13,30 m

The additional margins for sea level rise, land subsidence (0,1 m) and seiches (0,15 m) lead to a required crest level of MSL+13,55 meter. The present crest level is MSL+12.75 meter, which now seems to be insufficient for the moment.

Sophisticated procedure with regard to wave overtopping

The more sophisticated procedure using wave run-up formula (3) yields again different results. Using the same peak period again, the required crest level depends on the overtopping criterion.

Using formula (3) the required crest level is :

 0.1 l/m/s : hcrest = 4,70 m + 12,43 m = 17,13 m.

 1.0 l/m/s : hcrest = 4,70 m + 9,82 m = 14,52 m.

 10 l/m/s : hcrest = 4,70 m + 7,22 m = 11,92 m.

The additional margins for sea level rise, land subsidence (0,1 m) and seiches (0,15 m) lead to a required crest levels ranging from MSL+12,17 meter to MSL+17,38 meter. The present crest level is MSL+12.75 meter, which seems to fail the standards when 0.1 l/m/s or 1.0 l/m/s are allowed. The criterion of 10 l/m/s leads to an acceptable crest level.

5. Design of dunes

5.1

Introduction

In Error! Unknown switch argument. it is shown that the major part of the primary sea defence in the Netherlands consists of dunes.

The dunes have to protect the hinterland against flooding during severe attacks from the sea.

Error! Unknown switch argument. shows a typical Dutch dune cross section. Behind the rather slender dunes

frequently low-laying areas are found. Sometimes the ground level is even below mean sea level. Under normal conditions the hinterland is adequately protected by the dunes. During a storm (surge), however, the mean sea level rises and far higher waves approach the shore. Offshore transport, especially of material from the dunes, will occur. Error! Unknown switch argument. shows schematically what happens; sand eroded from the dunes is transported towards deeper water and settles there. After the storm the water level and the wave heights go to normal conditions again and the dune erosion process is stopped.

Figure Error! Unknown switch argument. Location of dunes in the Netherlands

Figure Error! Unknown switch argument. Typical Dutch dune cross section

In the Netherlands effort is put especially in judging whether existing dunes are safe or not under design conditions. To be able to judge the safety of a stretch of dunes it is necessary to predict the position of point R (see Error! Unknown switch argument.) after a storm surge. Taking into account the transfer function between the surge conditions and the amount of dune erosion, a probability of exceedance per year curve of the position of point R in the dune can be derived. Error! Unknown switch argument. shows an example for a very wide cross-section under Dutch conditions.

Figure Error! Unknown switch argument. Erosion as a function of time / frequency of exceedance

The actual design problem is in fact rather simple now. An acceptable pr obability of failure of the dunes has to be chosen (see paragraph 5.2). Based on probabilistic methods curves like the one in Error! Unknown switch

argument. have been derived for Dutch conditions. Those probabilistic methods are, however, rather laborious

for daily design purposes. That is the reason that in the Netherlands a rather simple design method, based on the results of extended probabilistic calculations, has been developed. This simple design method is

described in paragraph 5.3. An example of the safety assessment of a dune cross section is elaborated in paragraph 5.4.

5.2

Normative probability of failure

The design levels associated to the safety standards as determined by the Delta Commission (see paragraph 2) serve as a basis for the improvement of coastal dikes. Dikes in the Netherlands have to be designed in such a way that they can withstand a design storm surge. In such cases the dikes must still have some residual strength. Consequently, the frequency of exceedance of the design level may not be interpreted as a frequency of failure. The design method used in the Netherlands for dunes does not account for any residual strength. Consequently, dunes designed using this method should be able to withstand a storm surge with a lower probability of occurrence. The required safety margin during the occurrence of a water level equal to the design level is expressed by a factor with which the frequency of exceedance of the design level must be multiplied so as to arrive at a normative probability per year of collapse for a dune profile. This factor is set at 10-1. For Central Holland, for instance, this implies a normative probability of failure per year of 10-5.

5.3

Safety assessment of a cross section of a dune coast

5.3.1 Introduction

To evaluate the safety of dunes a simple safety assessment method described in the “Guide of the

assessment of the safety of dunes as a sea defence” is used together with a computational model to calculate the amount of erosion. The safety assessment method and the computational model are respectively

described in paragraph 5.3.2 and 5.3.3.

A dune is considered to be safe if a certain limit profile (see Error! Unknown switch argument.) is present after erosion has taken place under design conditions. If a dune is considered to be unsafe, a reinforcement

Figure Error! Unknown switch argument. Dune cross section

The actual amount of erosion during a storm surge is affected by the following 7 factors: 1. maximum storm surge level

2. significant wave height during the maximum of the surge 3. particle diameter of dune material

4. shape of the initial profile (including the dune height) 5. storm surge duration

6. occurrence of squall oscillations and gust bumps

7. accuracy of the computational model used to calculate the degree of dune erosion

The actual values of the parameters just before and during a certain surge are unknown. In a deterministic design method, design values of the several parameters have to be stated. If only a few parameters affect the result a deterministic design method seems appropriate. With at least 7 parameters it is, however, rather difficult to state beforehand reliable design values. With probabilistic methods, together with an appropriate selected chance of failure, a reliable design method can be derived. Usually the methods are discerned in levels:

Level III: Exact probabilistic methods taking into account the density functions of all parameters involved. Level II: Approximate methods in which the problem is linearized around a carefully selected point.

Level I: Design methods taking safety coefficients into account. Usually Level I methods are based on results of Level III or II calculations.

The method used in the Netherlands (see paragraph 5.3.2) can be seen as a Level I method. This relatively simple safety assessment method for the assessment of the safety of a cross section of a dune coast has been developed in such a way that the result corresponds with that of the more complicated probabilistic calculations.

5.3.2 The simple safety assessment method

The safety assessment method comprises a number of computational rules for the determination of that degree of dune erosion just before collapse. The values, to be used in the c alculations, are determined in such a way by probabilistic numerical techniques, that the thus calculated degree of dune erosion has a probability of exceedance equal to the required maximum accepted probability of collapse (see paragraph 5.2).

For some coastal sections, an additional amount of dune erosion, due to a gradient in the longshore transport, is still to be taken into account. In what way this is taken into account is not derived from the probabilistic calculations on which the safety assessment method is based, where this aspect was left aside (see paragraph 5.3.2.4).

The long-term development of a dune profile is of great importance, especially in case of an eroding coast. The safety assessment method has been developed in such a way that also a good impression can be obtained of the point in time when loss of the required safety of the dune profile might occur. Hence measures can be taken in time.

It is assumed that a series of profile measurements over the past 15 years or more is available. The yearly coastal measurements included in the data files of the automated processing system (Jarkus software) of the Rijkswaterstaat can be advantageously used. The availability of such a time series is not only imperative for

assessing the safety in the future, but also for the processing of the influence of the profile fluctuations on the safety. These fluctuations must be taken into account because it is not exactly known which profile is presen t just before the storm surge.

The procedure of the safety assessment method is as follows:

 An erosion analysis under design conditions is made for each profile from the series of profile measurements with the aid of the computational model described in paragraph 5.3.3. Specific

computational values need to be included here for the input parameters (storm surge level, significant wave height and grain diameter).

 For each erosion analysis, the calculated amount of dune erosion above storm surge level (A) is augmented with a surcharge (T) to take account of:

 the influences of the inaccuracy of the computational model;  the gust oscillations and gust surges;

 the uncertainty about the time during which the water level remains at about maximum level.  The effect of this surcharge is expressed in an additional recession of the steep dune front. Point P is the

intersection of this shifted dune front with the storm surge level (see Error! Unknown switch argument.).

Figure Error! Unknown switch argument. Definition sketch

 The above calculations yield a time series for the position of point P. These positions can be plotted in a diagram as a function of time (see Error! Unknown switch argument.). It can be easily induced from the position whether there is question of a stable, eroding, or progressing coast. The trend of the position of point P as a function of time can be estimated by means of regression analysis. A linear approximation will usually do. The profile fluctuations are expressed in the scattered position of the points P around this regression line.

Figure Error! Unknown switch argument. Principle of the safety assessment method

 The influence of the uncertainty of the profile position is now taken into account by shifting the regression line over a certain distance (d), dependent on the magnitude of the profile fluctuations, in a landward direction. The shifted regression line, the design erosion line, yields the position o f the design erosion point as a function of time. The design erosion point is the intersection of the steep dune front and the storm surge level here, the position of which has a probability of exceedance which is equal to the considered maximum permissible probability of collapse. In case of the straight coast of Central Holland, for instance, this probability is 10-5 per year. So far, the influence of a gradient in the longshore transport on the dune erosion has not been taken into consideration.

 For coastal profiles whereby account must be taken of the net loss of sand from the profile due to a gradient in the longshore transport, the final design erosion line is obtained by shifting the in the foregoing obtained shifted regression line over an additional distance ( g ) in landward direction.

 In case a minimum profile, the limit profile, no longer exists landwards of the design erosion line, the remaining profile no longer satisfies the established safety standard. Hence thi s limit profile does not offer a safety margin, but represents the situation just before collapse (limit state).

The following sections will discuss the foregoing in more detail.

5.3.2.1 The erosion analysis

With the aid of the computational model described in paragraph 5.3.3, an erosion analysis under design conditions is carried out for each profile from the available series of profile measurements. The following values must be used for the storm surge level, the significant wave hei ght, and the grain diameter of the dune sand:

The storm surge level

When assessing the safety in view of the function as a primary sea defence, the computational value for the storm surge level equals the design level, as established by the Delta Commission, plus a two third part of the decimation height. This level is called the computational level:

Computational level = design level* + 2/3 decimation height**

Hence the frequency of exceedance of the computational level is 0.215 times the frequency of exceedance of the design level and, consequently, 2.15 times larger than the concerned maximum permissible probability of collapse.

*

The design level has been defined as the (storm surge) level with (for Central Holland) a probability of exceedance of 10-4 per year. For Zeeland it is 2.5. 10-4 and for the Wadden Islands it is 5.10-4.

**

The decimation height is the difference in height between the water level with a probability of exceedance 10 times smaller than that of the design level, and the design level.

Table 1 represents the design level, the decimation height, and the computational level for a number of locations along the Dutch coast.

Table 1. Design levels, decimation heights, and computational levels along the Dutch coast

location design level (m above NAP) decimation height (m) computational level ** (m above NAP)

Vlissingen 5.40 0.72 5.90

Hoek van Holland* 5.25 0.72 5.75

Scheveningen 5.40 0.70 5.85 Katwijk 5.40 0.70 5.85 IJmuiden 5.15 0.67 5.60 Den Helder 5.05 0.66 5.50 Texel 4.90 0.68 5.35 Vlieland 4.70 0.68 5.15 Terschelling 4.80 0.68 5.25 Ameland 5.10 0.68 5.55

Schiermonnikoog 5.15 0.68 5.60

*

outside the breakwater

**

the computational levels are rounded off to a multiple of 5 cm

The significant wave height

The expected value of the wave height at computational level has to be used as the significant wave height Hs.

The probability density functions for the significant wave height as a function of the water level have been determined for a number of locations along the Dutch coast. The expected values of the significant wave height for these locations can be read from the diagram in Error! Unknown switch argument.. The given values hold for deep water conditions.

For a particular dune section, the expected value of the significant wave height at computational level can be found by means of this diagram.

Figure Error! Unknown switch argument. Expected value of the significant wave height as a function of the storm surge level at a number of locations along the Dutch coast

The grain diameter

The computational value (Dcomp) for the grain diameter is:

 

D









_

5

D50 = the expected value of the D50

D50 = the standard deviation of the D50

Representative values for a part of the sea strip along the Dutch coast are listed in Table 2.

Table2. Mean, standard deviation, and computational value of the grain size for a part of the sea strip along the Dutch coast.

location point of reference (km) D50 (m) D50 (m) Dcomp (m)

Schiermonnikoog 1.04 150 8 148

3.02 169 8 167

7.00 164 8 162

9.20 163 8 161

11.00 164 8 162

13.00 159 8 157

15.00 159 8 157

5.3.2.2 The surcharge on the amount of erosion above computational level

Three surcharges on the amount of dune erosion A (m3/m) above the computational level are to be included: - A surcharge of 0.10A (m3/m) to take into account the uncertainty about the time during which the water remains at about maximum level. This time span is the most determinative factor for the amount of dune erosion in the entire development of the water level during the storm surge.

- A surcharge of 0.05A (m3/m) to take into account the effect of gust surges and gust oscillations. - A surcharge of 0.10A + 20 (m3/m) to take into account the inaccuracy of the computational model for the expected dune erosion

The sum of the surcharges on the amount of dune erosion A above computational level consequently amounts to 0.25A+ 20 (m3/m). This surcharge is expressed as a landward shift of the originally calculated dune foot (see

Error! Unknown switch argument.).

5.3.2.3 Processing the profilefluctuations

The results of the calculations can be incorporated in a location -time diagram of the obtained point P (see

Error! Unknown switch argument.). A linear regression line for the position of point P in time can be

determined from this diagram, as well as the standard deviation of the position of the calculated points P fr om this line. The design erosion line is obtained by shifting this regression line landwards over a distance d:

d pz 2 275 where:

p = the standard deviation of the position of the calculated points P from the regression line (m)

z = mean value of the differences in height z between the most landward and the most seaward point of the total erosion profile of each erosion analysis (m) (see Error! Unknown switch argument.)

The constant in the denominator of the right term in the equation is determined in such a way that, using the safety assessment method, the desired outcome is obtained.

5.3.2.4 Processing a gradient in the longshore transport

In case of a varying longshore transport of sand along the coast (gradient in the longshore transport), for instance caused by obliquely approaching waves, an erosion -sedimentation balance for a particular coastal section does not exist. For safety reasons, those coastal sections are of importance where the

erosion-sedimentation balance has a negative outcome (total outgoing longshore transport exceeds the total incoming longshore transport).

The result is an additional landward shift of the erosion profile over such a distance that the cross -sectional area of the shift corresponds with the difference in longshore transport (see Error! Unknown switch

Figure Error! Unknown switch argument. The influence of a gradient in the longshore transport on dune erosion

A value of the gradient in the longshore transport due to a (not too strong) curvature of the coastline will be indicated in this section. Further investigations are required for strongly curved coastal sections, such as at the heads of islands. This also holds for other situations where a gradient in the longshore transport can be expected, such as at the transition between a dune and a structure (e.g. a breakwater, a dike, or a dune foot protection) and during strong variations in wave height in the along -shore direction (for instance behind sand-banks). The method is, therefore, inadequate for assessing the safety of such coastal sections. The value for the longshore transport G (m3/m) for not too strongly curved coastal sections, can be calculated with the formula:

G A Hs w G        * . . . . 300 7 6 0 0268 0 72 0 56 0 where:

A* = calculated amount of dune erosion above the computational level including the surcharge (m3/m) Hs = significant wave height at computational level (m)

w = fall velocity (m/s) G0 = reference value for G (m

3

/m) (see Table 3)

Coastal sections with a curvature according to class 1 are considered as straight coasts

(see Table 3). The entire coast from Hoek van Holland to Den Helder is considered to fall within class 1. The coastal curvature must be determined over stretches of at least several hundreds of metres. The final design erosion line for the concerned cross sections is obtained by shifting the

regression line (see Error! Unknown switch argument.) landwards over an additional distance ofg (m). Table3 Reference value for the difference in longshore transport for different classes of coastal curvature class curvature interval (degrees/1000m) G0 (m

3 /m) 1 0-6 0 2 6-12 50 3 12-18 75 4 18-24 100 5 >24 further invest

The distanceg is the mean value of the additional recession g, due to a gradient in the longshore transport, of the erosion point of each profile of the considered series of profile measurements (see Error! Unknown

5.3.2.5 The limit profile

The critical erosion point indicates the degree of dune erosion just before collapse. A minimum, yet stable profile (limit profile) must still be present landwards of the critical erosion point. The dune is supposed to collapse in case of a minor increase of dune erosion.

The dimensions of the limit profile are determined as follows:  The minimum crest level h0 is calculated with the formula:

h0CL012. T Hs

however,

h



CL



2 5

. ( )

m

Where:

CL = the computational level (m) above MSL T = peak period of the wave spectrum (s)

Hs = expected value of the significant wave height (m) at computational level

 The minimum width of the limit profile at crest level h0 is 3 m

 The gradient of the inner slope must be flatter than or equal to 1:2 The above is illustrated in Error! Unknown switch argument..

Figure Error! Unknown switch argument. The limit profile

5.3.3 Computational model / erosion profile

On the basis of extensive model investigations and prototype measurements, a computational model has been developed for the determination of the expected value and the standard deviation of the degree of dune erosion due to a random storm surge. The following factors are assumed to be known: the coastal profile before the storm surge, the grain size of the dune sand (D50), the storm surge level, and the significant wave

height. Starting points:

 the coastal profile is transformed into a certain erosion profile during a storm surge with dune erosion.  The shape of the erosion profile is a function of the significant wave height and fall velocity of the eroded

sand in still seawater.

 The shape of the erosion profile is independent of the angle of incidence of the waves, of the coastal profile before the storm surge, and of the storm surge level.

 It is assumed that the eroded sand is transported only in a seaward direction.

The erosion profile is situated in such a way with regard to the profile before the storm surge that the total area of the eroded sand is equal to the area of the deposited sand (see Error! Unknown switch argument.). It is generally assumed that no net loss of sand occurs in the alongshore direction.

During a storm surge the coastal profile is transformed into a certain erosion profile that is built up as follows:  After erosion has taken place, the dune foot, the point where the steep front of the eroded dune changes

into the relatively flat beach profile, is situated at storm su rge level. The gradient of the eroded dune slope is 1:1.

 Starting from the dune foot (x=0, y=0) in the seaward direction, perpendicular to the coast, the profile extents parabolically according to the formula:

7 6

0 4714

7 6

0 0268

18

2 00

.

.

.

.

.

H

y

H

w

x

















_



_





















Hs = significant wave height (m) in deep water

w = fall velocity of dune sand in seawater (m/s) x = distance to the new dune foot (m)

y = depth below storm surge level (m) to the point with co-ordinates:

x H w y H s s               250 7 6 0 0268 5 717 7 6 1 28 0 56 . . . . . .

Seaward form this point, the profile continues as a straight line with a gradient of 1:12.5 until it intersects the original profile.

The fall velocity w can be calculated with the formula:





10 1 10 2 10

0 476 2180 3 226

log . log . log .

w D D       where:

w = fall velocity of the dune sand in seawater (m/s) D = D50 of the dune sand (m)

5.4

Example

As an example of the safety assessment a dune cross section in Texel has been selected.

For this dike ring area or polder the safety standard is 1/4000 per year. The prescribed hydraulic boundary conditions for this location are:

 flood level: MSL + 4.30 meter  wave height: 9.35 meters  peak period: 12 seconds

An erosion analysis under design conditions is carried out for each profile from the available series of profile measurements (1985 to 1989). The following values have been used for the storm surge level, the significant wave height, and the grain diameter of the dune sand in the erosion analysis (s ee paragraph 5.3.2.1):  computational level = design level + 2/3 decimation height

= MSL +4.30 + 2/3*0.68 = 4.75 m  significant wave height at computational level = 9.35 m  computational value for the grain diameter = 179*10-6

m

The results of the erosion analysis are given in the following figures:

-5 -3 -1 1 3 5 7 9 11 -2 00 -1 00 0 ₁₀0 ₂₀0 ₃₀0 ₄₀0 ₅₀0

initial prof ile 1985 erosion prof ile

-5 -3 -1 1 3 5 7 9 11 -2 00 -1 00 0 ₁₀0 ₂₀0 ₃₀0 ₄₀0 ₅₀0

initial prof ile 1986 erosion prof ile

-5 -3 -1 1 3 5 7 9 11 -2 00 -1 00 0 ₁₀0 ₂₀0 ₃₀0 ₄₀0 ₅₀0

initial prof ile 1987 erosion prof ile

-5 -3 -1 1 3 5 7 9 11 -2 00 -1 00 0 ₁₀0 ₂₀0 ₃₀0 ₄₀0 ₅₀0

initial prof ile 1988 erosion prof ile

-5 -3 -1 1 3 5 7 9 11 -2 00 -1 00 0 ₁₀0 ₂₀0 ₃₀0 ₄₀0 ₅₀0

initial prof ile 1989 erosion prof ile

The amount of erosion A (m3/m) and the surcharge T (m3/m) (see Error! Unknown switch argument. and paragraph 5.3.2.2) is calculated and given in the following table for each of the five profiles:

profiles (year) A = calculated amount of dune erosion above computational level

(m3/m)

T = surcharge on A for:  duration of storm surge

 gust surges and gust oscillations  inaccuracy of the computational model (m3/m) 1985 20 25 1986 46 31.5 1987 79 39.75 1988 55 33.75 1989 63 35.75

The results of the calculations are incorporated in a location -time diagram of the obtained point P (see paragraph 5.3.2.3): -40 -35 -30 -25 -20 -15 -10 -5 0 1984 1985 1986 1987 1988 1989 1990 time (year) p o s it io n o f p o in t P

design erosion line regression line

A linear regression line for the position of point P in time can be determined from this diagram, as well as the standard deviation of the position of the calculated points P from this line. The design erosion line is obtained by shifting this regression line landwards over a distance d:

critical position

d

seaside

landside

d pz   2 2 275 5 45 12 18 275 13 . * . .

It is assumed that no net loss of sand occurs in the alongshore direction, whereasg is equal to zero (see

Error! Unknown switch argument. and paragraph 5.3.2.4).

The limit profile is calculated according to paragraph 5.3.2.5 and shown in the following figure:

-5 -3 -1 1 3 5 7 9 11 -2 0 0 -1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 initial profile 1989 limit profile

6. Developments

The situation described is largely based on the standard of practice in the Netherlands. The state of the art however shows a number of developments which may be introduced in the standard of practice in the near future. In some cases, mostly large scale flood protection projects like storm surge barriers, these

developments have been introduced already.

6.1 Probabilism

More probabilistic techniques

The Delta Commission introduced a one-dimensional probabilistic approach. The flood level was the only parameter considered to be a stochastic variable. The other parameters were treated in a deterministic way. In general, the other hydraulic loads like wind and waves were taken into account as expected values. At that time (1960) these expected values were ‘best’ or ‘educated’ guesses. The strength parameters were treated in a deterministic way too, but given the safety philosophy (to withstand the prescribed hydraulic loads) now conservative values or design criteria were used.

In the year following the report of the Delta Commission the hydraulic loads have been modelled in a more sophisticated manner :

 joint probability distributions of flood level, wave heights and wave periods have been derived for the coastal and lake areas

 joint probability distributions of flood level and wind velocities have been derived for the river deltas. The results of these studies are slowly but steadily introduced into practice. The safety standard (expressed as a return period) is applied to a combined hydraulic load parameter (e.g. overtopping discharge) instead of a flood level only.

The introduction of these developments however is always associated with a fierce discussion in which the focal issue seems to be : are we still in line with the principles of the Delta Commis sion?. In recent years it has been shown that the technical elaboration as mentioned above of the safety standards leads to higher

hydraulic loads, which again may lead to massive reconstruction works. On the other hand, probabilistic techniques are very much welcomed if traditional conservatism of certain design rules is replaced with a modern, but cheaper variety.

Uncertainties

A major issue in the discussion on probabilism is the way we deal with uncertainties. Uncertainties can be classified in three categories :

 implicit uncertainties, because the variable studies has a stochastic nature;

 model uncertainties, because our description of natural phenomena is always insufficient;  statistic uncertainties, because the number of observations of extreme events is too small.

The Delta Commission introduced the implicit uncertainties in our way of thinking. This has been extended in recent year to other hydraulic load parameters. Model uncertainties are not taken into account if hydraulic load models are cons idered. Strength models or design criteria do include a safety factor, although this factor is mostly based on experience or engineering judgement. Statistical uncertainties, like the accuracy of the design water levels (with a return period of 10.000 years) are not taken into account.

Some recent studies on uncertainties have shown that all uncertainties mentioned above can be incorporated into our design procedures. However, if these uncertainties are just treated as additional stochastic variables and the safety level is kept at the same level, this will lead to enormous increases in required crest levels. These increases may vary from 1.0 to 2.0 meters.

The example of the Pettemer Zeewering (paragraph 4.4) may be extended a little bit in order to explore the effect of additional stochastic variables and uncertainties. For this purpose several probabilistic calculations are made, according to the following assumptions. In all cases the sophisticated approach with res pect to wave run-up has been applied.

Scenario Water level Wave height Wave period Crest level

Reference (section 4.4) deterministic1 deterministic2 deterministic2 13,30 A) Stochastic water level stochastic deterministic2 deterministic2 13,36 A) Stochastic, uncertain water

level

stochastic uncertain3

deterministic2 deterministic2 13,87 A) Uncertain water level and

wave height

stochastic uncertain3

stochastic4 deterministic2 14,51 A) All hydraulic loads uncertain stochastic

uncertain3

stochastic4 stochastic4 15,32

1) Deterministic means that the water level with a return period of 10,000 years has been calculated separately. This value is used in a deterministic fashion to calculate the required crest level.

2) Deterministic means that the expected values of wave height and wave period are used to calculate the required crest level.

3) Stochastic and uncertain means that both the probability distribution function of the water level and its uncertainty are taken into account.

4) Stochastic means that the uncertainty of the expected values of wave height and wave period are taken into account.

The data used for the calculation is collected in appendix 1. As shown in the table the reference scenario leads to an almost identical crest level as the scenario A. This is also to be expected because the water level is the only stochastic variable in which case the design point for a deterministic calculation can be derived very easily. Increasing the number of stochastic variables however leads to increased crest levels :

 a statistic uncertainty ( = 0,0 and  = 0,35 meter) of the water level probability density distribution leads to an increase of the required crest level of approximately 0,50 meter;

 adding the uncertainty of the wave height ( = 1,0 and  = 0,20) leads to yet another increase of over 0,6 meter;

 finally, including the uncertainty of the wave period ( = 1,0 and  = 0,10) leads to the largest increase of over 0,80 meter.

Comparing with the reference scenario the added uncertainties leads to a total increase of the required crest level of 2 meter. 0 2 4 6 8 10 12 14 16 Reference A B C D C re st l e ve l [m ] -5% -1SD Mean +1SD +95% Design level

In figure 17 the results are shown graphically. This shows that the mean crest level is practically constant for all scenarios. Unfortunately the mean value is not significant for our purpose. The important extreme values however are very sensitive for the added stochastic variables and their statistical properties. The bandwidth between the extreme high and extreme low results increases dramatically as do the design levels which are related to a probability of 10-4 per year.

Basically for this reason the model (hydraulic) and statistical uncertainties are not yet taken into account in the standard of practice. The discussion on further application of probabilistic methods including uncertainties will be taken along in the research programme and policy preparation plan on flooding risks.

6.2 Safety philosophy based on flooding risks

The present safety standards are expressed as (return periods of) extreme water levels. These return periods and water levels are only indirectly related to the potential flooding risks, which were calculated in 1960. Technical uncertainties, like the behaviour of dikes during extreme conditions, prohibited a more direct link between economical damages or casualties and the technical requirements for flood protection structures. Since the report of the Delta Commission the issue of the calculation of flooding risk (probability of flooding times the consequences) has been a popular research item. Within this item many research topics can be discerned :

 geotechnical or structural modelling  strength parameters

 hydraulic modelling  hydraulic parameters

 statistical parameters, including correlation between various loads and events  modelling of failure or collapse of flood protection structures (breaching)  damages due to flooding

 effectiveness of measures to prevent damages.

At present the flooding probability (not the risk) has been calculated for a limited number of polders using state of the art modelling. Also an extensive uncertainty analysis will be carried out.

Polder Safety standard Flooding probability

Betuwe, Tieler- and Culemborgerwaarden (nr. 43) 1/1250 1/2000 Alblasserwaard and the Vijfheerenlanden (nr. 16) 1/2000 1/1100

Friesland and Groningen (nr. 6) 1/4000 1/3000

Hoekse Waard (nr. 21) 1/2000 1/1400

Terschelling (nr. 3) 1/2000 1/3000

Polder Mastenbroek (nr. 10) 1/2000 1/1400

Zuid-Holland (nr. 14) 1/10000 1/4000

Noord-Beveland (nr. 28) 1/4000 1/2000

In the following years flooding probabilities and flooding risks will be calculated for the entire country. These results can be used for the following purposes :

 to assess the actual flooding risks (damage potentials) related to the present safety standards;  to optimise the present design methods within the existing framework of safety standards;  to compare flooding risks with the societal risks associated to other events (e.g. traffic);

 to start a discussion on acceptable flooding risk levels in relation to acceptable risk levels of other events. The aim of this effort is to devise a new safety philosophy based on a flooding approach. With this, safety is related to the risk of flooding in terms of multiplying the probability of flooding with its consequences, expressed in damage and victims. This safety approach offers the possibility to consider and assess

measures in the entire risk chain (extreme water levels, the probability of a dike breach an d the consequences of flooding) and to make an optimal choice. Through measures which reduce the probability of high water or which limit the damage caused by a dike breach, just as great a contribution can be made to protection as with raising the height of the dike itself.

Error! Unknown switch argument. shows how the risk-concept and the regular safety assessment of dikes

other. The lower circle is the present policy of safety assessment aimed at maintaining the prescribed safety standard. The upper circle at the left side represents the future risk-assessment.

Figure Error! Unknown switch argument. Present flood protection policy and the risk-concept

The risk-assessment circle includes the socio-economic effects and the evaluation. This evaluation will not remain confined to flooding risks, other sources of risk will be taken into account as well. The risk -assessment will give information on expected damages in case of a flood. The damage of a flood by the sea will differ from the damage caused by water from the rivers: the water is or is not salt, announcement on long versus short term. A small polder will inundate more quickly than a large polder, through which the people get less time to evacuate. In a deep polder more damage will occur than in a shallow one. In a dike ring area where many people live and work, the damagelevel will be higher than in an area with thin po pulation. And last but not least, the damage depends on the people being prepared to evacuate, and how effective this evacuation takes place. The amount of damage may be accepted or rejected, given other sources of risk and the effort required to reduce the flooding risk. To reduce the flooding risk several strategies and measures can be considered. One of the alternatives is to heighten or strengthen the river dikes, which can be expressed as a higher safety standard. This safety standard can be maintained again using the lower circle, which is the core of the present flood protection policy. Given the time-scale of the processes involved the interaction between the both circles should not be frequent (safety assessment once in five years, safety philosophy once in 25 -50 years) .

6.3 Technical developments

Research is necessary to make the changeover from the current safety philosophy to the safety philosophy based on a flooding approach. The research programme Marsroute of the Technical Advisory Committee for Water Retaining Structures (TAW) aims to make this changeover possible.

In order to achieve an accurate safety philosophy based on the risk of flooding, it is essential that the probability of flooding and its consequences can be calculated sufficiently accurately. It is also important to establish what is felt to be an acceptable level of risk.

Prob ab ility of flooding

The probability of a dike breach is not adequately established under the current safety standards, while the probability of a dike breach followed by flooding is the most tangible measure of danger. After all, flooding results in economic damage and, depending on the situation, victims.

Merely measuring dike height provides insufficient information where it concerns protection from flooding. Two technical arguments can be cited for this: the geotechnical stability of dikes and the correlation between the failure of different dike sections

If for example, during periods of high water, the dikes lose their resistance to sliding, a dike brea ch can occur without the water flowing over the crest of the dikes. This contributes to the probability of a dike breach or flooding. The required resis tance to these largely geotechnical failure mechanisms cannot be expressed in terms of a hydraulic load standard or crest height. In the present situation, additional requirements are laid down for the probability of a dike breach occurring at water levels below the prescribed water level.

The larger the polder, the more dikes sections are needed to protect the area. If these sections together with the hydraulic load on these sections are fully correlated, the safety of the area can be expressed as the safety of a single dike section. In practice, this is not the case. Both the strength of and the load on the dike sections around the area are not fully correlated. Other types of constructions, such as discharge sluices, are to a large extent responsible for this. The probability of a dike breach in an arbitrary dike section, followed by flooding, is thus always greater than the probability of a breach in a single dike section.

Because of the variation in types of water-retaining structures, there is also a variation in the nature of the threats. After all, the threats to dunes are different to those to dikes. This means that the systems of dikes and water barriers can fail or collapse in different ways (mechanisms) and in different places (sections).The failure or collapse of a single element can cause the entire water-retaining system to fail, causing the area to become flooded anyway.

To calculate the probability of flooding, the Marsroute research concentrates on probability descriptions of load, the strength of the dikes and collapse mechanisms and the development of system failure models.

Consequences of flooding

Flooding usually results in extensive material damage. The extent of the damage depends on the nature of the threat (sea water or fresh water, short or long period of flooding, expected or unexpected) and the

characteristics of the flooded area (depth, built-up areas, industry, exact location of the dike breach). In particular, deep floods or fast-flowing water can have serious consequences in the form of victims, extensive damage, and disruption to normal life and infrastructure.

In calculating the consequences of flooding, the Marsroute research is concentrating on developing an instrument by which damage and victims for each dike ring area can be calculated in a uniform and practical manner. Warning and information systems contribute in taking the right measures at the right moment, by both government and the individual citizen. Applying these types of instruments influences the consequences of flooding in a positive way. A High Water Information System (HIS) is currently being developed for sections of the river lands. This can be used before and during high water for predicting the way in which any flooding will take place, monitoring water levels, waves, the condition of the dikes and the availability of the road network, determining the effects of any measures taken, announcements, communication and decision -making.

Acceptab le level of risk

The standards laid down in the Flood Protection Act are translated into standards based on the probability of flooding for each dike ring area. The current differentiation in standards for each dike ring area will mean other forms of safety in terms of flooding probabilities than in terms of overload frequency. The term "probability of flooding" is closer to the actual safety of an inhabitant of a dike ring area than the term "probability of overloading per dike section". A consequence of calculating the flooding probabilities will also be that a discussion will be initiated with regard to the greatest permitted level of the probability of flooding and its associated risk. Instruments for conducting this social discussion need to be developed.

Appendix 1 Data used for the probabilistic calculations

Water level

The probability density function used for the water level is an exponential distribution with the parameters 2,34 (shifted x-value) and 0,25 (-value). The uncertainty is described using an added variable with a normal distribution with the following parameters :  = 0,0 and  = 0,35 meter.

Wave height

The deterministic value used for the wave height is directly related to the water level. The figure below shows the relation between wave height (at MSL-20 metres) and water level. The wave height at the toe of the dike is reduced to a value of 4,70 meter for a surge level of MSL+4,70 meter. For this global calculations it is assumed that the wave height at the toe of the dike is generally equal to the surge level.

The uncertainty of the wave height used in probabilistic calculations is described using an extra variable with a normal distribution with the following parameters :  = 1,0 and  = 0,20.

Wave period

The deterministic value used for the wave period is 12,50 seconds. The uncertainty of the wave period used in probabilistic calculations is described using an extra variable with a normal dis tribution with the following parameters :  = 1,0 and  = 0,10.

Calculations

The probabilistic calculations are carried out using @Risk, which is a commercially available add -on for Excel or 1-2-3. This package features Monte Carlo or Latin Hypercube simulation techniques. For this case regular Monte Carlo simulation has been used with 10.000 simulations per run and for the final results 100 runs were executed. The design values presented (10-4 per year) were determined by averaging the results of these runs.