Based on the experimental results for profile change of rubble mound revetment, the modeling of damage progression process is discussed in this study. The ranks that indicate the degree of damage for the revetment are determined and the ratio for each rank of damage level is obtained from the experimental data. Three models based on Markov chain model are applied to the transition process of the ratio. The stationary transition probability is obtained for each model and the damage progression is predicted. The model that includes the transition to more than two ranks can predict the transition process of the ratio well

Hatono, A.

Matsumi, Y.

Ota, T.

Satow, T.

Hasanuddin University Repository

 332 
 
 
Proceedings of the 7th International Conference on Asian and Pacific Coasts  
(APAC 2013) Bali, Indonesia, September 24-26, 2013 
 
 
 
MODELING OF DAMAGE PROGRESSION FOR RUBBLE MOUND REVETMENT 
 
T. Ota 1, Y. Matsumi 1, A. Hatono1 and T. Satow 1 
 
 
ABSTRACT: Based on the experimental results for profile change of rubble mound revetment, the modeling of damage 
progression process is discussed in this study. The ranks that indicate the degree of damage for the revetment are 
determined and the ratio for each rank of damage level is obtained from the experimental data. Three models based on 
Markov chain model are applied to the transition process of the ratio. The stationary transition probability is obtained 
for each model and the damage progression is predicted. The model that includes the transition to more than two ranks 
can predict the transition process of the ratio well. 
 
Keywords: Damage progression process, markov chain model, maintenance 
 
 
                                                 
1 Dept. of Management of Social Systems and Civil Eng., Tottori University, 4-101 Koyama Minami Tottori, 680-8552, JAPAN 
INTRODUCTION 
The infrastructures are required to keep a certain 
level of performance during the in-service period and it 
is necessary to perform appropriate maintenance. 
Moreover, a model for deterioration process and repair 
of structures is needed for the efficient maintenance. A 
probabilistic model is especially needed for coastal 
structures that consist of many armor stones or blocks 
because of the uncertainty of deformation process. Satow 
et al. (2009, 2011) formulated a probabilistic model for 
preventive and corrective maintenance of armor blocks 
and examined an optimum maintenance policy 
theoretically. However, a small number of studies were 
conducted to investigate the deterioration process for the 
armor layer of coastal structures. As a step to develop a 
probabilistic model for the deterioration process and 
repair of rubble mound revetment, the modeling of 
damage progression process is discussed on the basis of 
the experimental results in this study. Specifically, the 
applicability of Markov chain model to the transition 
process of damage level is investigated. 
 
OVERVIEW OF EXPERIMENT TO MEASURE 
DAMAGED PROFILE 
The data of damage progression obtained from the 
previous laboratory experiments (Ota et al., 2011) are 
used in this study. The experiments to obtain many data 
of deformation quantity were conducted in 10 wave 
flumes that were set up in a multi-directional wave basin 
using the irregular waves. A conventional rubble mound 
revetment that consisted of a core and an armor layer 
was constructed on the flat bottom of each wave flume. 
The mean density and nominal diameter of the armor 
stone were s = 2.75 g/cm3 and Dn50 = 3.18 cm 
respectively. The mass of all stones was in the range of 
70 – 100 g. 
The JONSWAP spectrum with the shape parameter  
= 3.3 and significant wave period T1/3 = 1.2 s was used as 
the target spectrum of the incident irregular wave. The 
length of the input signal for wavemaker was about 27 
min and all wave paddles were driven by the signal in 
the same phase. The number of generated waves was 
approximately 1400 and the significant wave height H1/3 
was about 11 cm. The stability number based on the 
significant wave height, Ns = H1/3/(Dn50), was 2.0, 
where  = (sw)/w with w = water density. 
The profiles of each revetment were measured every 
27 min along three cross-shore lines using a laser 
displacement sensor at intervals of 1 cm horizontally. 
The normalized eroded area (damage level) 
2
50/ ne DAS   with Ae = eroded area was used as a 
parameter which showed the deformation quantity. The 
eroded area Ae was calculated using the damaged and 
initial (undamaged) profiles on each measurement line of 
the profile. The waves were generated in bursts of 27 
min repeatedly and the total time of wave generation was 
189 min (seven times) in a test except Test 1 where the 
total time was 162 min. The number of tests was 13 and 
the experiments were conducted under the same 
conditions of the incident wave and the initial profile of 
the revetment. A total of 2700 data of the revetment 
profile and the parameter S were obtained from the 
experiments. 
As a result of data analysis, the average Sm and 
standard deviation S of the parameter S for each number 
of iteration of wave action Nr were obtained as shown in 
Figure 1. Figure 2 shows the relation between Sm and S. 
In addition, the frequency distributions of S for each Nr 
 
Modeling of Damage Progression for Rubble Mound Revetment 
333 
 
were approximated by the normal distribution with Sm 
and S. 
A simple analysis was also made for the variation of 
damage level. The values of S were rounded off to the 
integers and the frequencies were obtained in all 
combinations of S on a same measurement line at 
adjacent Nr. The damage level was rated because the 
combinations of S were too many. Van der Meer (1987) 
proposed S = 2 for the start of damage and S = 8 for the 
failure of an armor layer on the 1:1.5 and 1:2 slope. 
Based on this criterion, the ranks of damage level were 
determined as follows; Rank a: S = 0 - 3, Rank b: S = 4 - 
7, Rank c: S = 8 - 11, Rank d: S = 12 or more. The 
relative frequency was computed for each combination 
of the rank as shown in Table 1. 
 
APPLICABILITY OF MARKOV CHAIN MODEL 
The relative frequency in Table 1 were obtained from 
the whole data of Nr = 1 - 7. The relative frequency for 
each rank of damage level at each Nr (hereafter, "ratio of 
damage level") is also obtained. Figure 3 shows the 
transition of the ratio of damage level. Because the 
damage levels of all revetments are zero at Nr = 0 that 
means the initial state, the ratio of Rank a is one. As Nr 
increases, the ratio of Rank a decreases rapidly and that 
of Rank d increases gradually. A few models using 
Markov chain with stationary transition probability is 
applied to the damage progression process in this study. 
The Markov chain model is often used for the 
deterioration prediction of the road pavement, bridges 
and concrete structures. The following three models are 
applied:  
Model 1; the damage level changes one rank at a time 
and all transition probabilities are same. 
Model 2; the damage level changes one rank at a time 
and the transition probability differs from each other. 
Model 3; the damage level can change more than one 
rank at a time and the transition probability differs 
from each other.  
The state transition diagrams of these models are shown 
in Figure 4 and the character 'p' in Figure 4 denotes the 
transition probability. Because the repair of the 
revetment is not considered in the present study, the 
damage level does not decrease. Rank d is the final 
phase of damage and the probability to stay in this rank 
is one. The ratio of damage level at Nr is given by Eqs. 
(1) - (3) for each model. 


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
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








0
0
0
1
100
010
001
0001 r
N
p
pp
pp
p
d
c
b
a
                       (1) 

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



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0
0
0
1
100
010
001
0001
3
32
21
1
rN
p
pp
pp
p
d
c
b
a
                  (2) 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 1 Average and standard deviation of S 
 
 
 
 
 
 
 
 
 
 
 
Fig. 2 Relation between Sm and S 
 
Table 1 Variation of damage level 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 3 Ratio of damage level 
 
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7
ra
ti
o
Nr
d
c
b
a
15
10
5
0
S
m
 ,
 
S
76543210
Nr
 S
m
 S
m
+
S
    S
m
-
S
4
3
2
1
0

S
121086420
Sm
0.218 0.642 0.134 0.006
0.563 0.416 0.021
0.731 0.269
1
a b c d
         N r +1
   N r
a
b
c
d
 
T. Ota, et al. 
334 
 
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
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0
0
0
1
1
01
00)(1
000)(1
342414
342313
242312
141312
rN
ppp
ppp
ppp
ppp
d
c
b
a
                
(3) 
The transition probabilities in the above equations are 
estimated by using Microsoft Excel 2007 (hereafter, 
"Excel"). The procedure is as follows (Port and Airport 
Research Institute, 2007). 
1) The initial values of the transition probabilities are 
arbitrarily given in the range of (0, 1) for each model. 
2) The ratios of damage level are calculated by Eqs. (1) - 
(3) for each Nr. 
3) The error sum of squares between the calculated and 
experimental values of the ratio of damage level at Nr 
= 7 is computed. 
4) The values of the transition probabilities that 
minimize the error sum of squares are calculated by 
using the Solver in Excel. The constraint condition in 
the parameter setting is that the all values of the 
transition probabilities are less than one. The addition 
item in the setting of option is the assumption of 
nonnegative number. 
The same estimated value of the transition probability p 
is obtained regardless of the initial value of p for Model 
1. The same estimated values of p1 - p3 are obtained for 
Model 2 except the case that all initial values of p1 - p3 
are 0.1. Because the estimated values of p12 - p34 for  
Model 3 are different depending on the initial values of 
them, the reasonable initial values are determined by 
reference to the result shown in Table 1. The estimated 
values of the stationary transition probabilities are shown 
in Table 2. Eventually, the error sum of squares at Nr = 7 
is about 0.03 for Model 1 and is on the order of 10-14 for 
Model 2 and 3. 
The ratios of damage level up to Nr = 20 are 
predicted using the estimated transition probabilities and 
Eqs. (1) - (3). To evaluate the predicted results by the 
Markov chain models, the ratios of damage level at Nr = 
8 - 20 are computed on the basis of the experimental 
 
 
 
 
 
 
 
(a) Model 1 
 
 
 
 
 
 
 
(b) Model 2 
 
 
 
 
 
 
 
 
 
(c) Model 3 
Fig. 4 State transition diagram for each model 
 
 
Table 2 Estimated transition probabilities 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 5 Approximation of Sm 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 6 Approximation of S 
 p p p 
1-p 1-p 1-p 1 
a b c d 
 p1 p2 p3 
1-p1 1-p2 1-p3 1 
a b c d 
 
p12 p23 p34 
1-(p12+ 
p13+ p14) 
1-(p23 
+p24) 
1-p34 1 
a b c d 
p13 p24 
p14 
20
15
10
5
0
S
m
20151050
Nr
 meas.
 pow er ap. of  S
m
 pow er ap. of  S
m
 exp. ap. of  S
m
Model 1 value Model 3 value
　p 0.3287 　  p 12 0.4045
Model 2   　p 13 0.0843
　p 1 0.4954   　p 14 0.0065
　p 2 0.3433 　  p 23 0.2729
　p 3 0.2075   　p 24 0.0370
　  p 34 0.1508
4
3
2
1
0

S
14121086420
Sm
 meas.
 pow er ap.
 Melby et al.
 
Modeling of Damage Progression for Rubble Mound Revetment 
335 
 
results as follows. The approximate formulas for the 
experimental values of Sm and S are expressed as Eqs. 
(4) and (5). 
     3588.0390.5 rm NS      (4) 
     4553.0038.1 mS S     (5) 
The solid lines in Figures 5 and 6 show the approximate 
curves. The dotted line in Figure 6 shows the empirical 
formula S = 0.5Sm0.65 that was given by Melby and 
Kobayashi (1998) based on the experimental result on 
rubble mound breakwater. By using Eqs. (4) and (5) to 
obtain the values of Sm and S at Nr = 8 - 20 and 
assuming that the probability distribution of S is given 
by the normal distribution with the average Sm and 
standard deviation S, the probability that the value of S 
falls within each Rank can be computed. In this study, 
the cumulative probability of the normal distribution 
P(S) is calculated by using the numerical integration by 
the Simpson's rule. Because P(0) is not zero and the sum 
of probability for each Rank is not one, P(0) is prorated 
according to the ratio of probability for each Rank. 
Figure 7 shows the ratios of damage level up to Nr = 20 
that are obtained by Model 1 – 3 and the normal 
distribution, however, the ratios up to Nr = 7 in (d) are 
the same as shown in Figure 3. These figures indicate 
that the ratio of Rank a decreases gradually and that of 
 
 
 
 
 
 
 
 
 
 
 
 
(a) Model 1                                                                              (b) Model 2 
 
 
 
 
 
 
 
 
 
 
 
 
(c) Model 3                                                                       (d) Normal distribution 
 Fig. 7 Ratio of damage level (models and normal distribution) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(a) Rank a, b                                                                              (b) Rank c, d 
Fig. 8 Variations of ratios by Rank 
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
ra
ti
o
Nr
d c b a
0
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d c b a
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d c b a
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ti
o
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d c b a
1.5
1.0
0.5
0.0
ra
ti
o
20151050
N
r
Rank a;  Model 1   Model 2, 3
             meas.+ nor.
Rank b;  Model 1   Model 2
             Model 3   meas.+ nor.
1.5
1.0
0.5
0.0
ra
ti
o
20151050
N
r
Rank c;  Model 1   Model 2
             Model 3   meas.+ nor.
Rank d;  Model 1   Model 2
             Model 3   meas.+ nor.
 
T. Ota, et al. 
336 
 
Rank d increases rapidly in the order, Model 1, 2, 3. 
Figure 8 shows the change of the ratios that are the same 
as shown in Figure 7 by Rank. Among the Markov chain 
models, the result of Model 3 shows the similar change 
of the ratio to that based on the experimental result and 
normal distribution. It is conceivable that the model 
including the transition to more than one rank can 
predict the transition process of the damage level. 
 
CASES THAT CHANGE CHARACTERISTICS OF 
Sm ARE DIFFERENT 
As described above, the ratio of damage level was 
computed by using the value of Sm shown by the solid 
line in Figure 5. Moreover, other processes of Sm are 
made by using other approximation for the experimental 
value of Sm and the ratio of damage level are computed 
to investigate its influence on the state transition. The 
filled circles in Figure 9 show the increments Sm that 
are obtained from the difference between Sm at adjacent 
Nr. The approximate curves and formulas are obtained as 
shown in Figure 9 by using the power and exponential 
approximation. The values of Sm at Nr = 1 - 20 can be 
obtained from these formulas and the dotted and broken 
lines in Figure 5 show the calculation results. The dotted 
line (power approximation) and broken line (exponential 
approximation) show a gradual increase and linear 
increase in Nr > 7 respectively. Because Melby and 
Kobayashi (1998) showed an experimental result that the 
average of S continued to increase with the cumulative 
number of waves, it is conceivable that the process of Sm 
with linear increase is possible. The value of S is given 
by Eq. (5), however, the use of S = 0.5Sm0.65 is also 
made in the case of exponential approximation. The 
probabilities that the value of S falls within Rank a, b, c 
and d respectively are computed by use of the normal 
distribution with above-mentioned Sm and S. Figure 10 
shows the ratios of Rank b, c and d at Nr = 8 – 20. In this 
figure, 'normal 1' is the same as the solid line in Figure 8, 
'normal 2' and 'normal3' are based on the power and 
exponential approximation of Sm respectively, and 
'normal 3+Melby' uses normal 3 and the empirical 
formula for S given by Melby and Kobayashi (1998). 
This figure shows that: 
- The gradients of decrease of Rank b and c and increase 
of Rank d correspond to those shown in Figure 5 at Nr 
= 8 - 20. 
- Although the change of Sm is same, the decrease of 
Rank b and c and the increase of Rank d are more rapid 
in the case of small S. 
- In the case of normal 3 that Sm increases linearly, the 
ratios of Rank c and d are similar to those of Model 2 
shown in Figure 8. 
 
CONCLUSIONS 
In this study, the damage level of rubble mound 
revetment was rated by the values of normalized eroded 
area S based on the experimental results and the ratio of 
the damage level and its transition were obtained. 
Furthermore, the applicability of three models using 
Markov chain with stationary transition probability to 
the change of the ratio was investigated. The transition 
probability for each model was obtained and the ratios of 
the damage level were estimated by using the probability. 
The results showed that the model including the 
transition to more than one ranks can predict the 
transition process of the damage level. The influence of 
the change characteristics of Sm on the state transition of 
 
 
 
 
 
 
 
 
 
 
Fig. 9 Approximation of Sm 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (a) Rank b 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (b) Rank c, d 
Fig. 10 Variations of ratios by Rank (normal 
distribution) 
 
6
4
2
0

S
m
20151050
Nr
S
m
 = 4.354 N
r
-1.136
S
m
 = 0.6569+18.67*
          exp(-1.393N
r 
)
0.3
0.2
0.1
0.0
ra
ti
o
2018161412108
N
r
rank b;  normal 1   normal 2
            normal 3   normal 3+Melby
1.5
1.0
0.5
0.0
ra
ti
o
2018161412108
N
r
rank c;  normal 1   normal 2
            normal 3   normal 3+ Melby
rank d;  normal 1   normal 2
            normal 3   normal 3+ Melby
 
Modeling of Damage Progression for Rubble Mound Revetment 
337 
 
the ratio was also examined. As a future work, a 
practical model including the repair of revetment will be 
made. 
 
REFERENCES 
Melby, J. A. and Kobayashi, N. (1998). Progression and 
variability of damage on rubble mound breakwaters. 
J. Waterw., Port, Coastal, and Ocean Eng., Vol. 124, 
No. 6, 286-294. 
Ota, T., Matsumi Y., Yamagichi, H. and Satow, T. 
(2011). Statistical characteristics of deformation 
quantity of rubble mound seawall, Coastal Structures 
2011. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Port and Airport Research Institute. (2007). Manual of 
maintenance technology for harbor facilities, 229p. 
(in Japanese) 
Satow, T., Matsumi, Y., Sumi, H., Hirayama, T. and Ota, 
T. (2009). A preventive maintenance problem of 
armour blocks with an in-service period. J. Coastal 
Eng., JSCE, Vol. 56, 951-955. (in Japanese) 
Satow, T., Matsumi, Y. and Ota, T. (2011). Discrete 
maintenance models of armour block facility under 
peaks-over- threshold, International Journal of 
Reliability, Quality and Safety Engineering, Vol. 18, 
No. 5, pp.483-494. 
 


Modeling of damage progression for rubble mound revetment

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Modeling of damage progression for rubble mound revetment

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