A coastal erosion model to predict shoreline changes

Measurement 47 (2014) 734–740

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A coastal erosion model to predict shoreline changes Francesco Adamo b, Claudio De Capua a, Pasquale Filianoti a, Anna Maria Lucia Lanzolla b, Rosario Morello a,⇑ a b

Department of Information Engineering, Infrastructure and Sustainable Energy, University ‘‘Mediterranea’’ of Reggio Calabria, Reggio Calabria, Italy Department of Electric and Information Engineering, Politecnico di Bari, Bari, Italy

a r t i c l e

i n f o

Article history: Available online 5 October 2013 Keywords: Coastal erosion Wave propagation direction Measurement uncertainty Shoreline prediction

a b s t r a c t Coastal erosion is a natural phenomenon affecting a growing number of worldwide sites. The impact of the waves on coast is cause of debris removal and soil erosion. The effect depends on wave strength, action time, and wave direction. In literature, several models have been proposed to estimate the mean rate of sediments moved annually alongshore. In the manuscript, the authors propose a prediction model to estimate the evolution of shoreline due to coastal erosion. Three altimeters are used to measure the instantaneous sea surface elevation. Directional wave spectrum is computed in order to estimate the direction of wave propagation and its measurement uncertainty. The shoreline is discretised into a finite number of linear segments. Then, according to historical information on the shoreline transformation, the impact of the wave on the coast is evaluated. Subsequently, the model predicts the changes of each line segment estimating the future shoreline. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Seaquakes, floods and coastal erosion are some effects of water impact on the soil. Such phenomena are due to large amounts of water colliding strikingly with earth. In such cases, water can be even cause of catastrophic events which shape the coast and the environment. The manuscript deals with coastal erosion issue and with prediction of shoreline changes. Such phenomenon is not caused only by sudden and extreme sea storms, but it is the consequence of the natural and continuous effect of the wave impacting against the coast. Consequently, attention is mainly turned on monitoring near-shore wave movements in sea and ocean. In literature, probabilistic models based on analyzing historical time series are widely used. Such solutions are cheap, however prediction results are often inaccurate and incomplete. Differently, coast monitoring ⇑ Corresponding author. Fax: +39 0965 875227. E-mail address: [email protected] (R. Morello). 0263-2241/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2013.09.048

systems are important tools in order to predict accurately planform and profile evolution of beach. So, real-time data concerning wave elevation and propagation have to be acquired. To this aim, near-shore sensor networks are commonly used to get timely measurements of the sea state [1–8]. Nevertheless such sensor networks suffer of synchronization problems [9,10]. In these cases, the wave impact on the coast is investigated by analyzing the directional wave [11]. Information on the direction of wave propagation is thus used to estimate the debris removal and the coastal erosion. About this issue, different models and procedures have been proposed in literature. For instance, surface buoys with mounted measurement equipment such as altimeters [1–4], and wave radar are frequently used for measuring wave elevation in nearshore and off-shore regions [12,13]. Even satellite image processing techniques based on segmentation algorithms have been proposed [14,15]. Nevertheless measurements are often inaccurate entailing false alarms or underestimations of hazard events (see Fig. 1).

F. Adamo et al. / Measurement 47 (2014) 734–740

735

Fig. 1. Wave elevation record.

The authors propose an innovative approach to predict the coastal erosion evolution. It is based on a model previously developed in [16], which is able to estimate the direction of wave propagation by means of elevation measurements. The direction of wave propagation is an important parameter to estimate the sediment transport. Measurement systems used in the practice are commonly based on wave-gauges or altimeters. Sensor networks or arrays are among the most widely proposed solutions for wave direction measurements. Buoy data are processed to get information about spectrum peak direction and incidence angle [17–19]. However, measurement uncertainty is often not considered during data processing. In the model proposed, the authors entail with this issue [16]. The directional wave spectrum is estimated to characterize wave movements and the sea state [20–24]. The impact of wave on the beach and the resultant erosion result depend on several factors such as the strength, the action time, the type of soil and mainly on the direction of propagation. Waves are mostly generated by the wind. In the beginning of generation, wave crests are very short and typically waves move in many directions. With the growth stage, waves propagate in the wind direction and the crests become longer. Wind waves typically leave the generation area becoming swells. The last ones cover long distances and are long-crested waves. In this sight, the direction of propagation of wind waves is easily estimated by considering the wind direction. Differently, more attention has to be paid on swells. So their representation in terms of directional spectrum is a basic approach to this issue. During motion, when wave is approaching to the coast, its energy disperses due to different factors such as sea depth variation or currents. Information on wave propagation angle is therefore used to optimize the prediction of the beach planform evolution. An innovative model starts from the current shoreline of the monitored coast. It is discretised into a finite number of linear segments. Then, according to historical information on the shoreline transformation and on the direction of wave propagation, the model predicts the changes of each line segment so to derive the next shoreline. A map of the beach profile evolution is depicted to compare the time trend and the level of coastal erosion risk. 2. Estimation of wave propagation direction Waves are principally originated by wind. The main factors which have influence on the size of wind waves are the wind speed and its duration, water depth, and the extent of sea surface affected by wind. The ideal sea state can be characterized by means of a sequence of wind waves in an undefined time interval with stationary state. Differ-

ently, the real sea state can be characterized by means of a discrete number of consecutive waves N  100–300. Such number is optimal to consider the state as stationary and representative of the sea conditions. With this assumption, the real sea state can be considered as a subsequence of the ideal one. Assume to consider a specific point of the sea. After a time, waves start to generate. They can be wind waves or swells. In detail, when the waves are directly due to local winds, we refer to wind waves. If the waves are not generated by local wind at that time, we refer to swells, so such waves have been generated elsewhere, or some time ago. Let us consider a buoy network in the sea being able to perform wave measurements at near-shore [1–4]. Each boy mounts on board an altimeter. So, we have to consider a record of the wave elevation g in the fixed point of the sea. This parameter changes with the time. So g(t) is the vertical displacement of the wave free surface referred to the undisturbed average level. Each wave can be characterized as the portion of g(t) between two consecutive zero upcrossings with the same slope. The period of the wave is the time interval between the two consecutive zero upcrossings. The crest and trough are the points on a wave with the maximum and minimum values respectively. The wave height is the vertical difference between the trough and the crest. The wavelength is the time interval from crest to crest. Suppose to consider several records gi(t) of the wave vertical displacement acquired by the buoy in the considered point of sea. Each record represents a real sea state with N waves. According to the first-order Stokes theory of sea state, the time series g1(t), g2(t), . . . , gn(t) are events of a stochastic stationary Gaussian process. Each event has an infinite time interval, so it represents an ideal sea state and can be described by the equation:

gðtÞ ¼

N X ai cosðxi t þ ei Þ

ð1Þ

i¼1

The ith element of the summation represents the vertical displacement of the wave free surface with amplitude ai, angular frequency xi and phase ei. The sea state theory assumes that:  N1;  ei values are stochastically independent and distributed over a round angle;  xi values are unequal each other;  ai values are infinitesimal. In order to estimate the line spectrum of the wave elevation, we have to consider time series g1, g2, . . . , gn with n being an odd number, where t1 = 0, t2 = Dtcamp, . . . , and tn = (n  1)Dtcamp. The Fourier series gF(t) is obtained by

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F. Adamo et al. / Measurement 47 (2014) 734–740

gF ðt1 Þ ¼ g1 ; . . . ; gF ðtn Þ ¼ gn N ¼ ðn  1Þ=2 2p i Dtcamp n n 2X g cosðxi tj Þ a0i ¼ n j¼1 j

xi ¼

a00i ¼

n 2X g sinðxi tj Þ n j¼1 j

and gF ðt þ T F Þ ¼ gF ðtÞ. The Fourier series gF(t) can be computed by the equation:

gF ðtÞ ¼

N X

ai cosðxi t þ ei Þ

ð3Þ

i¼1

Fig. 2. Direction of wave propagation.

Consequently the line spectrum EF(x) of gF(t) is given by [25–28]:

EF ðxÞ ¼

N X 1 i¼1

2

a2i dðx  xi Þ

ð4Þ

In order to analyse the direction of wave propagation, we have to estimate the directional wave spectrum starting from the previous equation. To this aim, the authors starts from the model in [29]. It was applied successfully also in [30] during a small scale field experiment carried out directly at sea. This model considers three buoys or measurements points A, B and C as in Fig. 2. In each point, an altimeter measures the wave elevation. According to the model in [29,30], the wave elevation in the three points can be estimated by the following equations:

gA ðtÞ ¼

N X

ai cosðxi t þ ei Þ

ð5Þ

ai cosðki Dx sin hi  xi t þ ei Þ

ð6Þ

i¼1

Fig. 3. Measurement uncertainty and direction of wave propagation.

gB ðtÞ ¼

N X i¼1

gF ðtÞ ¼

N X

a0i

cosðxi tÞ þ

a00i

sinðxi tÞ

ð2Þ

gC ðtÞ ¼

N X

i¼1

with

Fig. 4. The application case.

i¼1

ai cosðki Dy cos hi  xi t þ ei Þ

ð7Þ

F. Adamo et al. / Measurement 47 (2014) 734–740

737

Fig. 5. A specific portion of the beach profile.

where k is the number of waves, and hi is the angle of the direction of wave propagation. The previous equations can be rewritten in the following form:

gA ðtÞ ¼

N X A0i cos xi t þ A00i sin xi t

ð8Þ

i¼1

gB ðtÞ ¼

N X B0i cos xi t þ B00i sin xi t

ð9Þ

i¼1

gC ðtÞ ¼

N X C 0i cos xi t þ C 00i sin xi t

According to the linear theory of wind-generated waves in [29,30], the direction angle of the wave propagation hi is got by estimating the quantities A0i ; A00i ; B0i ; B00i ; C 0i ; C 00i . By resolving Eqs. (11) and (12) respectively, the solution is one if kiDx < p or as an alternative kiDy < p. Two solutions are possible if p < kiDx < 2p or as an alternative p < kiDy < 2p. We have more of one solution if kiDx > 2p or as an alternative kiDy > 2p. In brief, the solution for hi is unique if Dx and Dy are smaller than p/ki, so that

ð10Þ

i¼1

As a consequence, we can obtain the direction angle of the wave propagation:

tanðki Dx sin hi Þ ¼

A0i B00i  A00i B0i A0i B0i þ A00i B00i

ð11Þ

A0i C 00i  A00i C 0i A0i C 0i þ A00i C 00i

ð12Þ

or

tanðki Dy cos hi Þ ¼

sn Fig. 7. Coordinates of the pre-erosion ith linear segment.

si

s3 s2 s1 Fig. 6. Discretised shoreline.

Fig. 8. Coordinates of the post-erosion ith linear segment.

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F. Adamo et al. / Measurement 47 (2014) 734–740

Fig. 9. Prediction of the beach profile evolution.

A0 B00 A00 B0

hi ¼ arcsin

arctan Ai0 Bi0 þA00i B00i i i

i

i

ki Dx

ð13Þ

or A0 C 00 A00 C 0

hi ¼ arccos

arctan Ai0 C i0 þA00i C 00i i i

i

i

k i Dy

ð14Þ

The estimated direction of wave propagation is however affected by uncertainty. So the described model has been improved by considering the effect of the combined measurement uncertainty uc(hi). In detail, according to the Guide to the expression of Uncertainty in Measurement (GUM) in [31–37], if all input quantities are independent, the combined standard uncertainty can be evaluated by the expression:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi XM  @f 2 2 uc ðhi Þ ¼ u ðzj Þ j¼1 @z j

ð15Þ

where zj is the generic variable and M is the number of variables. As a consequence, the direction of wave propagation can change within an angular interval whose value is equal to hi ± uc(hi), see Fig. 3. The accurate estimation of the wave propagation angle allows us to optimize the analyse of the directional nearshore wave to predict reliably the beach profile evolution [38–40]. In detail, let us consider a specific site which is affected by coastal erosion, see Fig. 4. It represents the observed application case. In order to describe the proposed model, we have to consider the current shoreline of the beach. The aim is to predict the future shoreline by using the previous information on direction of wave propagation. So by way of example, in Fig. 5, the beach profile of a portion of the considered site is highlighted in red1 color. The shoreline is subsequently discretised into a finite number n of linear segments as in Fig. 6. Attention is now focused on the ith linear segment si. Let hi be the resultant direction of wave propagation esti1 For interpretation of color in Figs. 5 and 9, the reader is referred to the web version of this article.

mated by the previous equations, see green arrow in Fig. 6, while uc(hi) is its standard uncertainty. The resultant direction of wave propagation can be estimated by summing the direction vectors in proximity of the single shoreline segment. Once the resultant wave direction angle is estimated, we have to evaluate the wave impact on the single shoreline segment. Therefore, let us consider an orthogonal Cartesian system xy. Fig. 7 shows with more detail the ith linear segment and the associated wave direction vector, in black and green color respectively. Moreover, let be (xl,i; yl,i) the coordinates of the left segment extreme point and (xr,i; yr,i) the coordinates of the right segment extreme point, see Fig. 7 for reference. Starting from the information about the direction of wave propagation hi and the orientation of the ith shoreline segment, it is possible to estimate the relative orientation di of the wave with reference to the ith segment, see Fig. 8. According to historical information on the shoreline transformation for the considered zone, we can estimate the monthly average value D of the erosion effect. Such value depends mainly on the soil type and the wave strength impacting on beach. By means of these parameters, it is possible to estimate the average erosion effect and the debris removal. However it is not the aim of the present work, so we assume to know D. Consequently, the new coordinates of the segment extreme points can be estimated by the following equations:

x0l;i ¼ xl;i þ D  cosðdi Þ

ð16Þ

x0r;i ¼ xr;i þ D  cosðdi Þ

ð17Þ

y0l;i ¼ yl;i þ D  sinðdi Þ

ð18Þ

y0r;i ¼ yr;i þ D  sinðdi Þ

ð19Þ

Fig. 8 shows the segment translation due to coastal erosion effect, see halfway dash segment. According to information on standard uncertainty of the direction of wave propagation, we can define a variation range for the previous coordinates, see extreme dash segments. In this way, it is possible to estimate the deviation interval of the erosion effect due to measurement uncertainty. Since consecutive segments have one common extreme point, different wave vectors could insist on the same point, so the two erosion

F. Adamo et al. / Measurement 47 (2014) 734–740

effects have to be added. By applying the model to each segment, it is possible to predict the monthly changes of the whole shoreline and the consequent evolution of the beach profile. Simulations have been performed in order to estimate the shoreline changes of the considered site shown in Fig. 4. The model has been applied and the results are shown in Fig. 9. The current and the predicted shoreline profiles are shown in red and green color respectively. The results have shown a good agreement with the expected behavior. At the moment, experimentation is carrying out in order to validate the model. 3. Conclusions In the paper, the authors have proposed a model to predict the shoreline changes due to coastal erosion. The used procedure is based on a revised directional wave spectrum model previously developed by the authors in order to predict accurately the direction of wave propagation. Buoys with altimeters provide measures of the vertical displacement of the wave free surface. Information on wave elevation is then used to estimate the directional wave spectrum. Nevertheless, uncertainty affecting measurements can be cause of errors and inaccurate estimation of the wave propagation angle. Therefore, the measurement uncertainty is evaluated in order to optimize the estimation. The current shoreline of the monitored area is subsequently discretised into a finite number of linear segments, and the wave propagation angle is used to estimate the debris removal and so the translation of the single segment. A final map showing the evolution of the beach profile is depicted. The aim of the present work is to support the design of suitable intervention plans in the areas affected with greater risk of coastal erosion to reduce the consequent effects. References [1] M.L. Heron, W.J. Skirving, K.J. Michael, Short-wave ocean wave slope models for use in remote sensing data analysis, IEEE Transactions on Geoscience and Remote Sensing 44 (7) (2006) 1962–1973. [2] K.E. Steele, Pitch-roll buoy mean wave directions from heave acceleration, bow magnetism, and starboard magnetism, Ocean Engineering Journal 30 (2003) 2179–2199. [3] S.N. Londhe, Development of wave buoy network using soft computing techniques, in: Proc. IEEE OCEANS Conference 2008 – MTS/IEEE Kobe Techno-Ocean, 2008, pp. 1–8. [4] A. Sieber et al., ZigBee based buoy network platform for environmental monitoring and preservation: Temperature profiling for better understanding of Mucilage massive blooming, Proc. 2008 International IEEE Workshop on Intelligent Solutions in Embedded Systems (2008) 1–14. [5] L.F. Ferreira, P. Antunesa, F. Dominguesa, P.A. Silvaa, P.S. Andréa, Monitoring of sea bed level changes in nearshore regions using fiber optic sensors, Measurement 45 (6) (2012) 1527–1533. [6] k. Jinsoo et al., Accuracy evaluation of a smart phone-based technology for coastal monitoring, Measurement 46 (1) (2013) 233–248. [7] A. D’Orazio, A. Lay Ekuakille, C. Ciminelli, M. De Sario, V. Petruzzelli, F. Prudenzano, Architecture design of an integrated remote sensing center for now casting weather prediction inmediterranean area: an overview, in: Proc. European Geophysical Society Plinius Conference on Mediterranean Storms, 1999. [8] A. D’Orazio, C. Ciminelli, M. De Sario, A. Lay Ekuakille, V. Petruzzelli, F. Prudenzano, Genetic algorithm application for rass-wind profiler data processing in remote sensing, in: Proc. 2013 IEEE International

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