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WAVE BY WAVE OVERTOPPING ANALYSIS OF COASTAL STRUCTURES

João Miguel Figueiredo e Silva

Instituto Superior Técnico, Universidade de Lisboa

ABSTRACT

Coastal structures are very important for the Portuguese economy. These structures can be affected by

overtopping events, characterized by water mass transport over the crest of the structure. While

designing coastal structures, usually the mean discharge, qm [l/s/m], is the variable normally considered.

Recent studies have shown that the mean discharge fails to provide enough relevant information for the

design of coastal structures, and that maximum individual overtopping volume, Vmax [l/m], enables

additional relevant information.

By studying this phenomenon one is able to mitigate overtopping discharges, therefore reducing its

consequences.

A case study was considered in this paper: the final rehabilitation of the Sines harbor west breakwater.

This coastal structure suffered substancial damages in 1978-79 due to storms, which lead to mandatory

rehabilitation to reactivate the harbor. Experimental data was acquired at Laboratório Nacional de

Engenharia Civil (LNEC) from two dimensional scale physical model tests.

The present study follows a computational approach to obtain individual overtopping volumes from

measurements for a proposed solution. The mean discharges obtained using this methodology were

also compared to the mean discharges obtained in the previous studies.

To study individual overtopping volumes, a Weibull distribution was applied to data. The mean value

obtained for the Weibull’s β parameter was 0.83. No clear relationship was found between this

parameter and experimental wave characteristics.

INTRODUCTION

Nowadays, the evaluation of overtopping discharges in coastal structures is still done mainly using the

concept of mean discharge, q [l/s/m]. This variable has been related to the level of damage in

seawalls, buildings or other types of infrastructures and danger to pedestrians and vehicles (Pullen et

al., 2007). However, mean overtopping discharges alone may not give a complete overview of such a

dynamic and irregular phenomenon.

To have a more comprehensive evaluation of overtopping and the associated risks, it is important to

study also the wave by wave overtopping volumes, V [l/m] and estimate the maximum volume, Vmax.

Depending on the wave conditions and the structure type, the latter may be up to a hundred times

larger than q. Presently, there is already some guidance on tolerable values of Vmax (Pullen et al.,

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2007). Nevertheless, Vmax is calculated by methods that are not as well validated as those for the

determination of q and are limited to fewer types of structures. This study analyzes wave by wave

overtopping and mean overtopping discharges measured during physical model data collected at

LNEC for the West breakwater of Sines Harbour, Portugal.

The overtopping volumes per wave were calculated with both the time series of the surface elevation

at the back of the structure and with the time series of the water level variation inside an overtopping

tank. The number of overtopping waves, the individual volumes and the Vmax were the variables

estimated from raw data. The empirical probability distribution function of the overtopping volumes per

wave were compared with the Weibull distribution, a function often suggested in the literature for

rubble mound breakwaters (Van der Meer & Janssen, 1995).

WEIBULL EQUATION

Van der Meer & Jansse (1995) proposed a 2-parameter Weibull distribution:

�� = ��� ≥ ��� = � �− � �������

��� ( 1 )

P0 – Probability of exceedance of a given volume

Vi – Individual overtopping volume by unit length

β – Weibull configuration parameter

Vchar – Weibull scale parameter

����� = �, ���� = �, �� �������

( 2 )

� – Mean overtopping, assuming Weibull distribution

Pow – Wave overtopping probability

��� = !�! ( 3 )

N – Number of waves

Further studies shown that, if the number overtopping waves, N0, and the mean overtopping

discharge, qm, are known, the maximum overtopping volume can be determined by (Besley, 1999):

��� = ���" + �����$"%!�&'� ( 4 )

Vmin – Minimum observed volume

The average overtopping volume can be obtained using:

�� = �����( �'� + '� ( 5 )

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Where Γ is the gamma function.

CASE STUDY

Physical model tests of stability and overtopping were carried out in 2008 (Reis et al., 2011) in one of

the LNEC’s wave flumes to study solutions for the cross-sections of the final rehabilitation of the Sines

West breakwater. This study analyses both the mean and individual overtopping results for one of the

solutions (Figure 1).

Figure 1 - Studied cross-section in prototype values (Reis et al., 2011).

To determine the overtopping discharges in the flume, a tank was located at the back of each structure

to collect the overtopping water. The water was directed to the tank by means of a chute 50 cm wide

(Figure 2). A pump and a gauge were deployed in the overtopping tank and connected to a computer

that monitored and recorded the water level variation. Once a preset maximum water level was

reached in the tank the pump was activated for a fixed period. The pumped volume of water was

estimated from the pump calibration curve. The measurement of the water level variation inside the

tank, together with the pump calibration curve, allowed the determination of the mean overtopping

rates. To identify overtopping events and determine the wave by wave overtopping volumes a gauge

was located at the chute (Figure 2).

Figure 2 - Overtopping observed during physical model tests.

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For the solution studied on this work there where 24 tests with different water levels, peak periods and wave heights (Table 1). From those tests only 15 where considered, and they are marked as yellow.

Table 1 - Tests list

Prototype Model

Series Test Duration

(h)

Water

level

(m)

Peak period

(s)

Significant height Hs

(m)

Test length

(s)

Volume

(l)

Overtopping

(l/s/m)

A 1A 3 0 10 4 690 0,00 0,00E+00

2A 3 0 10 6 690 0,00 0,00E+00

3A 3 0 12 8 690 0,00 0,00E+00

4A 3 0 14 9 690 0,00 0,00E+00

5A 6 0 16 10 2760 15,17 1,10E-02

6A 6 0 18 11 2760 55,65 4,03E-02

7A 6 0 18 11,5 2760 74,31 5,38E-02

8A 6 0 12 12 2760 0,39 2,83E-04

9A 6 0 16 12 2760 45,06 3,27E-02

10A 6 0 20 12 2760 180,86 1,31E-01

11A 6 0 20 13 2760 480,80 3,48E-01

12A 6 0 20 14 2760 590,13 4,28E-01

B 1B 3 4 10 4 1380 0,00 0,00E+00

2B 3 4 10 6 1380 0,00 0,00E+00

3B 3 4 12 8 1380 0,00 0,00E+00

4B 3 4 14 9 1380 9,63 1,40E-02

5B 3 4 16 10 1380 64,57 9,36E-02

6B 6 4 18 11 2760 436,74 3,16E-01

7B 6 4 18 11,5 2760 743,91 5,39E-01

8B 6 4 12 12 2760 28,73 2,08E-02

9B 6 4 16 12 2760 480,47 3,48E-01

10B 6 4 20 12 2760 1042,30 7,55E-01

11B 6 4 20 13 2760 1940,37 1,41E+00

12B 6 4 20 14 2760 2800,60 2,03E+00

On the yellow marked tests the bomb was not used on test 5A, 8A, 3B and 4B.

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DATA PROCESSING

The data processing was made using Matlab®. The program had some extra parameters that might

change depending on the test (Table 2). Some sensibility tests where made to evaluate how those

parameters would affect the overall results and to fit them to the specific data.

Table 2 - Specific test parameters.

RUNAVF DELA SPAR SUBS FILMF SDMF AVPSD PERC SPANA MARGIN LOOKMAX LOOKMIN

5A 0,4 0,6 0,4 0,3 10 5 1 0,9 0,2 1 1 1

6A 0,4 0,4 0,6 0,3 10 5 1 0,75 0,2 0,8 1 0,4

8A 0,9 0,6 0,6 1 10 4,1 1 0,9 0,2 1 1 1

9A 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 0,1 2 2

10A 0,4 0,4 0,6 0,3 10 5 1 0,85 0,2 0,7 1 1

11A 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 1,1 0,3 0,4

12A 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 0 1 1,2

3B 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 1 1 1

4B 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 1 1 1

6B 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 0 1 1

8B 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 1 1 1

9B 0,4 0,4 0,6 0,3 10 5 1 0,8 0,2 0 1 1

10B 0,4 0,4 0,6 0,3 10 5 1 0,9 0,2 0 1,5 2

11B 0,4 0,4 0,6 0,3 10 5 1 0,8 0,2 0 1 2

12B 0,4 0,4 0,6 0,3 10 5 1 0,8 0,2 0 1,2 1

After the raw data and the parameter file are loaded, the program starts by converting the data from

volt to liters using the tank calibration line (Figure 3).

Figure 3 – Tank calibration line.

y = 0,0442x2 + 7,5669x + 0,0466R² = 1

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6

Tan

k V

olum

e (l

)

Water surface reading (V)

Curva decalibração

Calibration

line

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While doing the conversion the program verifies if the tank reaches its limit and erases the data in

which the pump is working and connects the data after that to the last point before the pump starts

working again. By following this procedure, the program artificially creates an infinite tank (Figure 4,

left panel)

Figure 4 – On the left, test 6B raw data converted to liters. On the right is presented the data in liters

without the pump with the stacking volume.

After the tank level data is processed to be read by the program it’s time to read data from the crest

probe. The data, in Volts, is used to detect overtopping events, but it does not detect every event, in

most cases. A filter, FILMF, was designed to distinguish between noise and overtopping (Figure 5).

Figure 5 – Crest probe data with noise filter.

The program determines the time for each overtopping event detected using the filter. Since there is a

delay between an overtopping detection and an increase in tank level, a delay is applied to get the

correct time on tank data.

Some data files present noise caused by external sources. This noise can be mitigated using a moving

average to better detect events on tank data. Using these detected events detected, mean and

standard deviation parameters are calculated between each event. The program searches for the first

point bigger than the average and calculates a threshold based on it to set if there is a new

overtopping event. If a new event is detected it’s stored and the same applies between this new point

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and the previous point until no more new events are detected, or the data ends.

To determine the volume for each event an average is calculated between each event, the

corresponding overtopping volume to an event is the calculated subtracting the average right after the

event by the average right before the event.

In this last part there is an extra verification to make sure there are no miscalculated events. If any

event has overtopping below zero the event is omitted and every overtopping volume is recalculated

(Figure 6).

Figure 6 – Individual overtopping volume representation.

RESULTS

The sum of all individual volumes obtained using the program was compared against previous studies

(Table 3).

Table 3 - Comparison between individual overtopping obtained and previous studies

Test Measured volume (l) Program volume (l) Diference (l) Diference (%)

5A 15,17 10,90 4,27 28,16

6A 55,65 46,77 8,88 15,96

8A 0,39 0,18 0,21 53,26

9A 45,06 38,99 6,07 13,47

10A 180,86 168,74 12,12 6,70

11A 480,8 340,63 140,17 29,15

12A 590,13 367,86 222,27 37,66

3B 0 0,00 0,00 0

4B 9,63 9,63 0,00 0,05

6B 436,74 272,90 163,84 37,51

8B 28,73 28,88 -0,15 -0,53

9B 480,47 307,92 172,55 35,91

10B 1042,3 741,88 300,42 28,82

11B 1940,37 1166,25 774,12 39,90

12B 2800,6 1312,43 1488,17 53,14

The difference between the two studies increases for bigger volumes. There is also a low overtopping

test that had shown a huge difference, but eye check on this test shows some problems on raw data.

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Figure 7 – Individual overtopping volume representation

There is also a comparison between mean discharges (Table 4).

Table 4 - Mean discharge comparison

Ensaio qm medido (l/s/m) qm programa (l/s/m) Diferença (l/s/m) Diferença (%)

5A 5,11 3,67 1,44 28,18

6A 18,74 15,75 2,99 15,96

8A 0,13 0,06 0,07 53,85

9A 15,18 13,13 2,05 13,50

10A 60,91 56,83 4,08 6,70

11A 161,93 114,72 47,21 29,15

12A 198,75 123,89 74,86 37,67

3B 0,00 0,00 0,00 0,00

4B 6,49 6,48 0,01 0,15

6B 147,09 91,91 55,18 37,51

8B 9,68 9,73 -0,05 -0,52

9B 161,81 103,70 58,11 35,91

10B 351,03 249,85 101,18 28,82

11B 653,49 392,77 260,72 39,90

12B 943,20 442,01 501,19 53,14

The number of overtopping events in each phase of the program (Table 5).

Table 5 - Number of overtopping events

Overtopping

events. First

interation

Overtopping

events. Second

iteration

Final number of

overtopping

events

5A 13 247 31

6A 29 216 83

8A 23 29 11

9A 27 53 53

10A 60 156 156

11A 78 139 139

12A 77 183 149

3B 0 0 0

4B 11 19 19

6B 76 151 151

8B 16 28 28

9B 82 239 201

10B 87 200 200

11B 79 192 190

12B 81 222 185

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Based on the previous data the Weibull distribution was fitted to the data as shown in Figure 8.

Figure 8 - Overtopping observed during physical model tests.

Applying equation 2 and 3 to the Weibull distribution results, it was possible to estimate the average

and maximum overtopping values (Table 6)

Table 6 - Measured maximum and average overtopping volume compared with maximum and average volume obtained for both regressions using the program.

Test Average

volume

(l/m)

Average volume

Weibull (LS)

(l/m)

Average volume

Weibull (LAD)

(l/m)

Maximum volume

(l/m)

Maximum volume

Weibull (LS)

(l/m)

Maximum volume

Weibull (LAD)

(l/m)

5A 42,19 48,63 48,95 191,54 243,69 248,41

6A 67,61 60,83 63,62 635,89 394,81 443,63

8A 1,99 2,03 2,01 8,80 5,00 4,99

9A 88,28 89,60 86,67 435,18 358,99 338,19

10A 129,80 121,87 135,04 879,68 859,86 1093,57

11A 294,07 290,12 305,00 1361,78 1478,01 1696,85

12A 296,27 273,71 287,14 2125,14 1441,47 1742,78

4B 60,79 64,56 67,77 248,30 205,10 216,58

6B 216,88 207,40 219,47 1580,04 1163,71 1386,83

8B 123,78 87,61 80,87 1746,49 337,47 330,08

9B 183,83 189,10 191,58 1782,68 1658,40 1691,33

10B 445,13 469,38 500,30 1990,76 2933,15 3390,20

11B 736,58 803,71 773,15 3746,09 4277,12 3695,38

12B 851,31 947,86 850,81 7015,80 6204,64 4906,82

The obtained average volume is similar to the data for both regressions.

Regarding the maximum overtopping volume, in both regressions, the have similar values, but they in

most of the cases far from the data.

CONCLUSIONS

The wave-structure interaction is characterized by complex physical phenomena, including

overtopping. The overtopping phenomenon is an important factor to consider in choosing the crest

level. This affects the safety of the surrounding area.

This study used the Matlab® to evaluate the overtopping phenomenon. A Weibull distribution was

applied to the data obtained during physical tests applied to a scale model.

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The program is able to estimate the individual overtopping based on the raw data. Using this individual

overtopping data, it computes the Weibull distribution parameters, which enables a comparison to be

made with the structures characteristics.

This study allowed one to conclude that:

• The Weibull distribution describes the overtopping phenomenon with a correlation coefficient

of at least 0,95, for the considered data;

• The β parameter of the Weibull distribution for this structure is 0,86 for a Least Square

regression and 0,83 for a Least Absolute Deviation regression. Both of them are close to the

values found in previous studies (β = 0,75);

• The Least Absolute Deviation method is a better regression technique to describe this

phenomenon, since it’s more robust;

• There was not a clear relationship between β and wave characteristics.

• The average volume obtained using a Weibull distribution is near the average volume on the

experimental data, but that is not true for the maximum overtopping volume.

It is recommended that the number of tests without pump should be increased. To increase the

number of tests it’s advised to use this program to Solution 1 and 3, since this study was only meant to

test Solution 2. It’s also advised to make improvement on data acquisition, so there are fewer

problems with the raw data. This will lead to better adjustments in the program and consequently to a

better understanding of the overtopping phenomenon.

ACKNOWLEDGEMENTS

The author gratefully acknowledges the Sines Port Authority for the permission to use and publish

physical model data.

REFERENCES

Besley, P. (1999). Overtopping of Seawalls. R&D Technical Report W178 & W5/006/5, Environment

Agency, Bristol, UK.

Pullen, T., Allsop, N.W.H., Bruce, T., Kortenhaus, A., Schuttrumpf, H., Van der Meer, J.W., 2007. Eurotop:

Wave Overtopping of Sea Defenses and Related Structures: Assessment Manual, Environment

Agency, UK, Expertise Netwerk Waterkeren, NL, and Kuratorium fur Forschung im

Kusteningenieurwesen, DE, August, 178p.

Reis, M.T., Neves, M.G., Lopes, M.R., Keming, H., Silva, L.G., 2011. Rehabilitation of Sines West

Breakwater: Wave Overtopping Study. Maritime Engineering, 164, 1, 15-32.

Van der Meer, J., Jansen, J., 1995. Wave Run-up and Wave Overtopping at Dikes. In: Wave Forces on

Inclined and Vertical Wall Structures, New York, USA Task Committee on Forces on Inclined and

Vertical Wall Structures, ASCE, pp. 1-27.