at SciVerse ScienceDirect

Ocean & Coastal Management 69 (2012) 102e110

Contents lists available

Ocean & Coastal Management

journal homepage: www.elsevier .com/locate/ocecoaman

A comparison between three short-term shoreline prediction models

Rodrigo Mikosz Goncalves a, Joseph L. Awange b,*, Claudia Pereira Krueger c, Bernhard Heck d,Leandro dos Santos Coelho e

aDepartment of Cartography Engineering, Federal University of Pernambuco (UFPE), Geodetic Science and Technology of Geoinformation Post Graduation Program,Recife 50670-901, PE, BrazilbDepartment of Spatial Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, AustraliacGeodetic Science Post Graduation Program, Federal University of Parana (UFPR), Box 19.001, 81.531-990 Curitiba, PR, BrazildGeodetic Institute, Karlsruhe Institute of Technology, Engler-Strasse 7, D-76131 Karlsruhe, Germanye Pontifícal Catholic University of Parana, PUCPR Production and Systems Engineering Graduate Program, LAS/PPGEPS Imaculada Conceicao, 1155, 80215-901 Curitiba, PR, Brazil

a r t i c l e i n f o

Article history:Available online 23 August 2012

* Corresponding author.E-mail addresses: rodrigo.mikosz@ufpe.br (R.

curtin.edu.au (J.L. Awange), ckrueger@ufpr.br (C(B. Heck), leandro.coelho@pucpr.br (L.dosS. Coelho).

0964-5691/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.ocecoaman.2012.07.024

a b s t r a c t

Monitoring and management of shorelines along populated coastal areas is a very important task, butremains a difficult endeavor. The historical information used for short-term analysis and prediction arealways underpinned by uncertainties associated with old data. Predictions of shoreline positions nor-mally depend on the accuracy of the input data as well as the validity of the mathematical models used.With the requirement to study shoreline changes along the Parana (PR) coast in Brazil, it was necessaryto obtain related cartographic information, which included temporal shoreline data obtained fromorthophotos. In this contribution, photogrammetric together with GPS data are used to compare thecapability of three shoreline prediction models; linear regression, robust parameter estimation, and neuralnetwork to predict the 2008 Parana shoreline position, which is then validated using the GPS measuredposition of 2008. The results indicate a MAPE (Mean Absolute Percentage Error) of 0.61% for the linearregression, 0.14% for the robust estimation, and 0.33% for the artificial neural network method. Althoughthe coefficient of determinant (R2) value for the neural network was the best, i.e., 0.997 compared to0.994 for the robust model and 0.984 for the linear regression, its maximum deviation from the controlvalues (i.e., 16.46) was almost twice that of robust model (7.63). On the one hand, the robust estimationmodel provides a more suitable approach for managing outliers in shoreline prediction, and also vali-dating traditional methods such as linear regression. On the other hand, the neural network methodoffers an alternative approach to the robust prediction model. The results of the study highlighttheimportance of a model choice for predicting the shoreline position.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Most coastal areas are known to experience erosion. In Brazil,for example, hundreds of beaches are known to suffer severeerosion problems (e.g., Souza, 2009). Beaches are sources ofrevenue for many countries through tourism, and as such, neces-sitate proper management in coastal areas. For the efficientmanagement of beaches, however, one of the important tasksremains that of accurately monitoring the coastal shorelines.Gorman et al. (1998) presented an overview of the methods of datacollection for coastal monitoring and baseline studies using

M. Goncalves), J.Awange@.P. Krueger), heck@kit.edu

All rights reserved.

satellite images, aerial photography, profile surveys and bathy-metric records. Remote sensing is important for creating base mapsable to interpret landform changes and quantify shoreline move-ment. The hydrographic and also large-scale topographic maps areconsidered the primary sources of information for shoreline posi-tions and volumetric change computations. The use of profilesurveys can also contribute to evaluating shoreline changes andcomputing beach volume changes along and across the shore.

The importance of coastal monitoring information, e.g., on thestate of the current erosion and the possible prediction of its impacthave been shown by Li et al. (1998) to be critical to decision-makingin coastal management. Li et al. (1998) point to the fact that coastalmonitoring information could be useful in land-use zoning,construction setbacks, and relocation implementations. They givea practical example of a digital topographical map that is overlaidwith a cadastral map and an erosion condition map, which is used to

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110 103

find lots and parcels that are affected by coastal erosion. Anotherarea that requires shoreline monitoring is that of the integratedcoastal zone management (ICZM), where such information assist inthe process of coastal management and territorial planning/decision-making by providing essential knowledge about coastalprocesses and their dynamic evolution (e.g., Veloso-Gomes et al.,2008). ICZM has been shown to be important for example in thedetection of landscape and vegetation changes, and to establish therelations between human impact and vegetation in natural coastaldune systems (see, e.g., Tzatzanis et al., 2003).

Monitoring of beach stability, therefore, is vital for environ-mental as well as resource management, and is essential forimproving databases of information on shoreline evolution in anarea. The Metropolitan Borough of Sefton (2002) listed the benefitsof shoreline evolution information as; providing input to shorelinereview plans, planned maintenance of coastal defenses, achievementof high government level targets, determination of appropriate designcriteria for coastal works, biodiversity action plans, implementation ofhabitats directive, and leisure and amenity management of shorelineareas. While shoreline monitoring, as discussed above is essential,predicting its future position is equally vital to support coastalmanagement and impact assessment programs, e.g., predicting thenecessary future building setbacks from the shoreline to serve asprotection for a time comparable to the expected lifetime of newcoastal structures, usually 30 or 60 years (Hecky et al., 1984;Crowell et al., 1997). Crowell et al. (1997) point out, however, thatdetermining adequate setbacks require estimating long-termshoreline change trends from historical data. For historical data-set, metric quality and aerial photographs are very important andare some of the most reliable data sets available for coastal zones.Crowell et al. (1997) and Douglas et al. (1998) reckon that often thedata used in such predictions are at times temporally poorlysampled historical databases. They point out that short records arenot good predictors of shoreline location.

Fenster et al. (1993) developed a predictive method that detectsshort-term changes in the long-term trend and identifies linear orhigh-order polynomial models that best fit the data according tothe minimum description length (MDL) criterion. In this method,only linear models are extrapolated. Predictions shaped or influ-enced by higher-order polynomial schemes can sometimes besuperior to those obtained from linear regressions, but they canalso be extremely inaccurate (Crowell et al., 1997). Douglas et al.(1998) stressed the need to incorporate long-term erosion trendsand historical record of storms, including their impacts on shore-line positions and beach recovery in predictivemodels. Thieler et al.(2000) tested the use of mathematical models to predict beachbehavior for U.S. and cited lack of hind-sighting and objectiveevaluation of beach behavior predictions for engineering projects,and incorrect use of model calibration as some of the motivationbehind their work.

Shorelines are known not to be stable and vary over time. Short-term changes occur over decadal time scales, or less, and are relatedto daily, monthly, and seasonal variations in tides, currents, waveclimate, episodic events and anthropogenic factors (e.g., Demarestand Leatherman, 1985; Galgano et al., 1998; Galgano and Douglas,2000). Fenster et al. (2000) describe shoreline movement asa complex phenomenon and outline the difficulties involved indistinguishing long-term shoreline movement (signal) from short-term changes (noise).

Even with the presence of noise in the data, the use of modernaccurate surveying techniques has been shown to improve thequality of forecasts. For example, Douglas and Crowell (2000)demonstrated that meaningful deduction was achievable even ifthe inherent variability of shoreline position indicators remained atthe level of many meters. Other than the surveying technique used

for observation, a challenge for detecting, monitoring, and pre-dicting shoreline movement is to develop an effective mathematicaltool capable of identifying and mapping a feature set as “shoreline”according to the data sources available (Boak and Turner, 2005). Forinstance, exploiting the relationship between shoreline and sealevel changes (i.e., using series of sparsely sampled sea-level valuesas surrogate data for shoreline change), Douglas et al. (1998)developed an algorithm that evaluates some of the predictivemethods such as the end point method, linear regression, andminimum description length criterion and established that linearregression gave superior results. The need for an effective mathe-matical tool is further emphasized by Li et al. (2001) who stress theneed to integrate specific coastal engineering and modeling soft-ware packages in a GIS environment since they are usually notprovided by commercial GIS software system.

The problemwith the linear regressionmodels is that they workwell when the underlying linearity and normal distributionassumptions are fulfilled. However, in some cases, where the datamaybe of poor quality, the linear regression assumptions may beviolated. Uncertainties in the extracted shoreline data need to beappropriately addressed if they are to be used to predict futureshoreline positions to support sustainable coastal management(see, e.g., Addo et al., 2008). Robust estimation models have beenproposed to deal with cases where data contains uncertainties (see,e.g., Huber, 1964, 1981; and Hampel et al., 1986).

Using the positions of shorelines over time, inferred fromphotogrammetric and GPS surveys for the years 2001, 2002, 2005and 2008, this study discusses the possibility of developing a short-term prediction model using different sources of temporal geodeticdata, withmost of them having different accuracies. To achieve this,a comparison and assessment of three different shoreline predic-tion models; linear regression, robust parameter estimation, andartificial neural network is undertaken.

The robust estimation model used in this work has beensuccessfully applied by Awange and Aduol (1999) to estimategeodetic parameters in the case of contaminated observations. Theartificial neural network method tested in this paper is termedNeural Network Multilayer Perception (MLP), and was imple-mented using the training of LevenbergeMarquardt algorithm(Hagan and Menhaj, 1994).

The study is organized as follows. In Sect. 2, the background,study area, data, and the shoreline prediction models are discussed.The results are then presented and discussed in Sect. 3, and thestudy concluded in Sect. 4.

2. Data and methods

2.1. Background

Since 1996, the Spatial Geodesy Laboratory and Hydrography(LAGEH) at the Federal University of Parana (UFPR) have beencollecting data using GPS surveys over the Parana State coast (see,e.g. Krueger et al., 2009). The data are composed of digital carto-graphic documents with reports showing the post-processing andaccuracy reached in a specific survey. With a proposal to studyshoreline changes along the Parana coast, some analog images/photographs related to the years 1954, 1963, 1980, 1991 and 1997were retrieved and used for shoreline extraction along a 6 kmcoastal zone of the Matinhos beaches in the state of Parana, Brazil.

2.2. Study area

The study area is located in theMatinhos District along the coastof Parana State, Brazil (Fig. 1), where the beaches (105 km long)form the second smallest littoral along the Brazilian coast, located

Fig. 1. Matinhos District at the coast of Parana State, Brazil.

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110104

between latitudes 25�S and 26�S and longitudes 48�W and 49�W.In geological terms, Soares et al. (1995) describe the area as beingcomposed of Pre-Cambrian units of crystalline complexes ofigneous and metamorphic rocks. Pleistocene and Holocene sandysediments of marine origin from a terrace (beach ridges) are foundbetween the shoreline and the foot of the Serra do Mar Mountains.The beaches of Matinhos are subject to both oceanic conditions andebb tidal delta influence from the neighboring Guaratuba Bay(micro-tidal system). The district of Matinhos, with an area of117 km2, comprised 29,172 habitants in 2010. Land use is mostly forrecreational purposes and in summer, the city becomes moredensely populated due to the influx of tourists.

The settlement in Matinhos is largely near the shore, wheresettlements are characterized by constructions at backshore posi-tions or over the beach leading to the destruction of dunes andwetlands, thereby forcing rivers to change course. This was a resultof lack of urban planning and the fact that the morphology andcoastal dynamic environment was not considered during thesettlements. Fig. 2 shows the situation in 2007 and 2008 at the

Fig. 2. State of the Matinhos beach in

study area, indicating the prevalence of the problems of coastalerosion.

2.3. Data source

Monitoring of shoreline positions nowadays benefit from thestate of the art mapping techniques such as global navigationsatellite systems (GNSS, e.g., Goncalves, 2010; Awange, 2012),remote sensing using satellite images, aerial photographs (e.g.Mitishita and Kirchner, 2000), and LIDAR (Light Detection andRanging). For instance, Mitishita and Kirchner (2000) extractedshoreline position from temporal vertical aerial photographs by themonorestitution technique, which required altimetry informationand was dependent on the quality of the photos, the distributionand number of control points, and photo-interpretation for shore-line extraction.

In the present study, aerial photographs and GPS observationscollected for a 6 km part of the coastal zone were used, where thephotogrammetric data was converted to digital orthophotos using

2007 and 2008 (Goncalves, 2010).

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110 105

control points and digital terrain model (DTM). The shorelinepositions were then obtained using the monorestitution techniquebased on photo interpretation. The GPS data was collected duringa series of geodetic surveys of the shoreline in Matinhos using thekinematic relative positioning method (see, e.g., Awange, 2012).The GPS survey was undertaken by LAGEH (Laboratory of SpaceGeodesy and Hydrography) in UFPR (Federal University of Parana)using dual frequency GPS receivers. The base station was installedat Pedra located at 25� 490 5.779900S; 48� 310 49.136400 W inMatinhos.

All temporal vectors of shorelines from photogrammetry for theyears 1954,1963,1980,1991 and 1997 (Fig. 3a) and fromGPS for theyears 2001, 2002, 2005 and 2008 (see Fig. 3b) were placed in layersand processed in the same geodetic reference frame (WorldGeodetic System; WGS-84). For distance computations, planarcoordinates (UTM) were used. The shoreline positions obtainedfrom photogrammetric data did not always begin and end at thesame location, and also had gaps, and as such did not cover all theinformation at the study site.

2.4. Shoreline prediction models

Inwhat follows, brief discussions of the linear regression, robustestimation, and artificial neural network models are presented.

2.4.1. Linear regression modelThe linear regression model commonly used in shoreline

prediction is expressed as

y ¼ axþ c1; (1)

Fig. 3. Temporal resolution of the p

where x and y are the linear regression vectors, while a and c arethe unknown linear regression coefficient and intercept respec-tively, and 1 is a column vector consisting of “1” numbers. Equation(1) can be expressed in terms of the observations and unknowns as

y ¼ Abþ ε; (2)

with y being an n � 1 column vector of observations, A is an n � mdesign matrix (herem¼ 2), b is them� 1 vector of unknown linearregression parameters, and ℇ is the n � 1 vector of observationalerrors. Equation (2) can now be expressed in the form of theGausseMarkov estimation model as:

y ¼ Abþ ε; εw�0; s20W

�1�; (3)

whose solution (linear regression) is given by

bb ¼�ATWA

��1ATWy; (4)

with the dispersion Dfbbg ¼ bs20ðATWAÞ�1 andbs2

0 ¼ ðεTWεÞ=ðn�mÞ. Here, bs20 is the variance of unit weight and

W is the n � n positive-definite weight matrix.First, we define a reference line from which cross-sections are

taken at intervals of 100m. The temporal positions of shorelines arethen obtained from their intersection with the cross-sections.These positions are then used to calculate the shoreline distancesxn, related to time tn, which forms the elements of the observationvector y in equation (1). The temporal information, i.e. years, areused in the design matrix A.

The linear regression solution (4) holds when there are no grosserrors and bb becomes an unbiased estimator of b since the

hotogrammetric and GPS data.

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110106

expectation Efbbg ¼ b. With gross errors in the observations,however, we have

y ¼ Efyg þ 2þ ε; (5)

where2 is the n � 1 bias vector on the observational vector y.Considering y ¼ y þ dy, the solution of (5) from (4) leads to:

bb ¼�ATWA

��1ATWy þ

�ATWA

��1ATWdy; (6)

with the expectation

E�bb� ¼

�ATWA

��1ATWy þ

�ATWA

��1ATW2 ¼ bþ g ¼ b0;

(7)

and y ¼ Efyg; dy ¼ 2þ ε: It becomes evident then that bb isa biased estimator of b with the termðATWAÞ�1ATW2. Settingg ¼ �db, equation (7) leads to

b ¼ b0 þ db: (8)

2.4.2. Robust estimation modelThe robust estimation model is defined as follows: Consider

x1,x2,.,xn as being randomly independent variables assumed tohave a normal distribution described by the probability densityfunction F0(x). If a fraction k(0 < k < 1) of the observations arecontaminated by gross errors, then the new distribution will berepresented by Huber (1964):

FðxÞ ¼ ð1� kÞF0ðxÞ þ kHðxÞ; (9)

where H(x) is the unknown contaminating distribution. The robustestimation procedure estimates parameters from this setup in sucha way that the influences of the gross errors in the final estimatedparameters are significantly reduced.

In adopting a robust estimation procedure, we seek a rigorousadjustment method that will solve equation (6) and give the resultsthat are as close as possible to those of equation (4). One advantageof using robust estimationmethods is that they do not immediatelydelete outlying observations from a given set, rather, they isolatetheir effects through, e.g., down-weighting. The iterative weightingapproach proposed by Aduol (1994), for instance, provides a meansof down-weighting bad observations.

In this approach, the weights of observations are considered asfunctions of both the observational variances and the observationalresiduals. In equation (3), the weight matrix is defined as

W ¼ S�1; (10)

with S being the covariance matrix of the observations. This weightis modified in the robust iterative weighting approach such thatequation (10) takes into consideration the residuals. Equation (10)is thus re-written as

W ¼�R2 þ S

��1; (11)

where

R ¼

2664n1 0 : 00 n2 : 0: : : :0 0 : nn

3775; (12)

and n1, n2,., nn are the observational residuals from equation (3)expressed as

v ¼ y � Abb: (13)

The values of v obtained from equation (13) are re-introducedinto equation (11) to obtain a new weight matrix W. The proce-dure for the estimation of bb is then repeated until a sufficient levelof convergence is attained. In this iterative approach, the startinginitial values of the weight matrix W are those obtained from thelinear regression solution in equation (4).

This method makes use of the fact that any gross error in theobservation will manifest itself in the estimated residuals, whichwill tend to be larger compared to the others due to the effect of thegross errors. In the re-estimation of the elements of the weightmatrix W, a relatively large residual will result in a relatively lowweight for the respective observation. The procedure thereforedown-weights the influence of outlying observations and in sodoing, reduces the effect of bad observations on the estimatedquantities.

2.4.3. Neural networkIn beach morphodynamics applications, Álvarez-Ellacuría et al.

(2011) have shown using an example that artificial neuralnetworks can be used to obtain shoreline positions from daily rawvideo images where the spatial and temporal variability of thebeach are split by Empirical Orhthogonal Function (EOF) decom-position. For the setup and development of the artificial neuralnetwork (NN), the characteristics of neurons, topology, and trainingrules must be specified. The adaptation of the initial weights andlearning of their behaviour are specified by the rules of training.The algorithms of training of a NN have the feature of adjustingiteratively the weights of connections between neurons until thepair of inputs and expected outputs are obtained, and consequentlythe relations of cause and effect established. When the setting ofa particular problem presented to NNs changes, and the perfor-mance model does not fit the situation anymore, it is possible totrain the NN with new conditions of input and output to improvethe performance (Kröse and Smagt, 1996; Haykin, 1999).

The schematic model of an artificial neuron considered as theprocessor of a neural network was proposed by McCullochePitts,where three basic elements can be identified (see, e.g. Kröse andSmagt, 1996; Haykin, 1999; Arbib, 2003). The first element is a setof synapses or connecting links, each characterized by a weightwkj.The second element relates to a sum of the input signals, weightedby the synapses of the neuron (linear combination) as

vkXmj¼0

wkj$xj: (14)

The third element is a transfer function which limits the outputamplitude of a neuron to a finite value, given by

yk ¼ 4ðnkÞ: (15)

A neuron can be described mathematically using equations (14)and (15), where 4 represents the transfer function of the artificialneuron, which processes the set of inputs received, while changesin the activation state are required to obtain a good fit or model forthe problem at hand. The log-sigmoid transfer function (Equation(16)) can take values between 0 and 1, where a is the slopeparameter of the sigmoid function and n is the activation of theneuron.

f ðnÞ ¼ 11þ eð�anÞ (16)

Equation (17) is the hyperbolic tangent function, which takesvalues between 1 and �1, where a is the slope parameter of the

Table 1Artificial neural network settings.

Architecture e Multilayer Feed-Forward Networks e Structure

Artificial neural network MLPTraining method LevenbergeMarquardt(LM)Number of hidden neuron 1Number of hidden layer 4Activation function used in hidden layer Hyperbolic tangentActivation function used in output layer Linear

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110 107

curve; b is the lower and higher limiting values and n is the value ofactivation.

f ðnÞ ¼ aeðbnÞ � eð�bnÞ

eðbnÞ þ eðbnÞ(17)

The linear activation function is defined by Equation (18) wherea is a real number that defines the linear output to the input values,y is the output and x is the input.

y ¼ ax (18)

An important concept of a NN is the definition of the architec-ture, or the way in which neurons in a network may be arranged,which is an important parameter that restricts the type of problemthat can be treated in a specific network. Networks with a singlelayer of nodes, for example, can only solve linearly separableproblems. Recurrent networks, in turn, are more appropriate tosolve problems that involve temporal processing (Haykin, 1999).

The Multilayer Feed-Forward Network was used with theLevenbergeMarquardt (LM) algorithm for training in this work.This type of architecture is composed of an input layer of source,one or more layers of hidden neurons and another layer of outputneurons. There are several types of methods for training networksthat are grouped into two categories: supervised learning andunsupervised learning. The case of supervised learning or learningwith a teacher has a set of inputeoutput examples. However in thecase of learning without a teacher (unsupervised learning), there isneither teacher to oversee the learning process nor are therelabeled examples of the function to be learned by the network (see,e.g., Haykin, 1999).

The most popular algorithm of error management is the backpropagation, were the idea is to correct the errors according toa learning rule. The learning consists of two steps through thedifferent layers of the net according to their direction; one stepforward, the propagation, and one step backward, the back prop-agation (e.g. Kröse and Smagt, 1996; Haykin, 1999; Arbib, 2003).

In the forward step, one vector of input is applied to thesensorial nodes and it in effect propagates through the net, layer bylayer. This results in one set of output to be produced with a realanswer or estimate by the net. During the propagation through thenet, all weights are fixed. During the backward step, all weights areadjusted according to a rule of correction of errors (see, e.g. Kröseand Smagt, 1996; Haykin, 1999; Arbib, 2003). Specifically, theanswer of the net is subtracted from that desired to produce theerror signal. This signal is propagated back through the net, hencethe name back propagation. The weights are adjusted to make theanswer of the real net closer to the desired output (see, e.g. Kröseand Smagt, 1996; Haykin, 1999; Arbib, 2003).

In practice, the use of the back propagation algorithm tends toconverge very slowly, requiring a great computational effort. Tosolve this problem, several techniques have been incorporated toimprove the performance in order to reduce the convergence timeand the computational effort required. Such techniques include,e.g., the LevenbergeMarquardt algorithm (see, e.g. Hagan andMenhaj, 1994; Lera and Pinzolas, 2002).

A test using neural networks is done with the characteristic ofnon-supervised training. The inputs are the vectors of observationsy described in equation (2). The network learns the relationshipsprovided by the input information and finds out what the outputfor the forward step representing the 2008 predictions are. Thedata for the year 2008 does not participate in the training. Similarlyto the linear regression and robust estimation methods, the 2008data serves only as a control to verify the answers found by therespective predictive model. It is emphasized that any kind ofweight is selected as the input data for the neural network case.

Table 1 show the setting used to perform the neural network. It isimportant to report that we tried many more neural networkconfigurations with different input and hidden layers (or neurons)to check the performance of each model, with the best one selectedand compared to the other methods.

2.5. Evaluation criteria

Three experiments were performed; the first with the linearregression model (equation (4)), the second with the robust esti-mation model (equations (10)e(13)), and the third with the neuralnetwork (Table 1). To assess the efficiency of the three methods, wecomputed the following:

(1) The mean of the deviation from the GPS 2008 derived shore-line, which we used as a control. This is computed using

x ¼ 1n

Xni¼1

ðGPS 2008i � Prediction 2008iÞ; (19)

where (GPS 2008i � Prediction 2008i) denotes the distancebetween the predicted shoreline and the control shoreline at thecross-section, while i and n are the number of sections.

(2) The standard deviation is then obtained from the square root ofthe average variance of the estimated shoreline position, i.e.,

s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

ðGPS 2008i � Prediction 2008i � xÞ2vuut (20)

(3) The root mean square (RMS) of the shoreline deviation iscomputed to gain a measure of the corresponding effectivenessof the method, i.e.,

RMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

ðGPS 2008i � Prediction 2008iÞ2vuut : (21)

(4) The coefficient of determinant R2 was computed to assess thefit of the predicted value to the true shoreline position using(see, e.g., Schaible and Lee, 1996).

R2 ¼ 1�Pn

i¼1ðGPS 2008i � PredictioniÞ2Pni¼1ðGPS 2008i �Mean GPS 2008Þ2

(22)

(5) Finally, we determine the mean absolute percentage error(MAPE), which measures the accuracy of a fitted time seriesand usually expresses accuracy as a percentage (see, e.g., Hayatiand Shirvany, 2007).

Table 2Cross-sections characteristics.

Cross-sections Temporal data belonging to the set Degrees offreedom

10 1963 2001 2002 2005 215 1963 1980 2001 2002 2005 310 1954 1963 1980 2001 2002 2005 47 1954 1963 1980 1991 2001 2002 2005 59 1954 1963 1980 1991 1997 2001 2002 2005 61 1963 1980 1997 2001 2002 2005 45 1963 1980 1997 2001 2002 3Total ¼ 57

Table 3Results of the prediction models for the 2008 shoreline position compared to that of the

Sample number Number of iteration forrobust best solution

Robust (GPS2008 � Prediction)

Absolute value (m)

1 12 0.852 13 1.363 28 1.014 1 1.315 14 0.96 3 0.937 1 0.298 20 0.479 2 2.1310 2 0.4511 27 2.2412 2 0.6813 2 0.8114 2 1.5815 2 0.2316 2 1.7617 2 0.2318 1 1.6119 1 0.7920 1 1.6021 1 0.7822 1 1.2423 1 0.3224 1 1.2925 1 1.2426 2 0.0627 21 0.5028 4 0.4529 4 0.7030 3 0.1231 3 0.6932 2 1.1833 2 3.0134 2 2.1535 5 0.3036 7 6.0037 19 0.3638 3 0.3639 1 1.5740 3 0.0741 1 4.2142 3 1.4643 1 4.7744 5 0.0145 1 6.7346 1 6.2147 1 0.7348 3 0.2849 2 1.9750 2 0.2851 2 0.6852 1 7.6353 2 2.3654 1 0.2755 3 0.0456 1 1.2057 7 6.20

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110108

MAPE ¼ 100n

Xni¼1

����GPS 2008i � Prediction 2008iGPS 2008i

���� (23)

2.6. The 2008 predictions

All historical data are placed in layers and once this is done, oneis chosen as a shoreline reference upon which the cross-sectionsare drawn. In this case, the first line obtained by GPS (2001) wasselected as a reference, since it comprised the limits of extension of

GPS survey in 2008 (GPS 2008-model results).

Linear regression (GPS2008 � Prediction) Artificial neural network(GPS2008 � Prediction)

Absolute value (m) Absolute value (m)

14.35 3.5414.54 3.898.94 2.061.31 2.335.29 1.486.48 1.350.29 0.197.58 0.032.20 4.624.75 3.30

12.84 4.946.54 0.346.62 0.313.97 0.885.23 1.054.04 0.685.35 1.411.61 0.730.79 1.291.60 0.120.78 0.791.24 1.040.32 0.911.29 1.961.24 1.783.12 7.861.85 3.983.08 3.843.28 2.660.36 4.281.44 5.383.70 0.548.57 2.619.22 0.25

28.74 2.6257.29 12.0947.36 14.1513.68 16.461.57 2.325.37 1.884.21 2.834.51 2.304.77 4.961.99 1.266.73 11.816.21 11.960.73 3.433.23 1.182.10 3.141.36 2.353.28 3.057.63 13.572.62 12.510.27 4.351.77 2.451.20 1.527.00 3.33

Table 4Statistical data.

Statistics results Robustmodel (m)

Linearregression (m)

Neuralnetwork (m)

Maximum 7.63 57.29 16.46Minimum 0.01 0.27 0.03Mean 1.56 6.45 3.69Standard

deviation1.83 10.1 4.03

RMS 2.39 11.9 5.36R2 0.994 0.984 0.997MAPE 0.14 0.61 0.33

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110 109

the coastline studied. From it, cross-sections with a distance of100 m were drawn.

In practice there are 57 transverse lines and Table 2 presents therespective information; number of cross-sections, temporal infor-mation of the data, and the degrees of freedom relative to the linearmathematical model, i.e., the difference between variables and thenumber of independent equations.

For the weight matrix in equation (10), the photogrammetrydata are assigned a standard error of 16 m, i.e. 2s following Crowellet al. (1991) who suggests an error of about s¼ 8m. GPSmeasuringaccuracy of 2 cm is adopted for the kinematic relative GPS data (see,e.g., Awange, 2012). With this information, the vector of theunknowns comprising the linear regression parameters a and c isestimated. The established linear regression models are then usedto predict the shoreline position for the year 2008. This procedure isrepeated with the robust estimation and the neural networkmethods. The results of the predicted shoreline for 2008 from allthe three methods under study are compared with that of the 2008GPS survey.

3. Results and discussion

Table 3 shows the results for the three prediction methods:linear regression, robust estimation and neural networks,compared with the true values based on the 2008 GPS survey usedas the control for the set of 57 samples used in the study. Thesecond column of Table 3 shows the number of iterations per-formed to obtain the best solution of the robust estimation. Whenthe number of iterations is 1, the solution found by the robustestimation method is essentially the same as that of the linearregression. As can be seen from the results of Table 3, Robustestimation method has the smallest values of the absolute errors,i.e., GPS position of 2008 e the predicted model positions. This isvisible in Fig. 4, which presents the deviation of the predicted 2008shoreline position from the GPS control data of 2008 constructedfrom the data in Table 3. The absolute errors are indicated in y-axis(m) and the samples appear on the x-axis.

In Fig. 4, the results of the 37th sample are seen to have thelargest absolute error as clearly depicted by the results of theRobust estimationmethod. This sample could possibly be an outlierdue to the uncertainty in its data. The advantage of the Robustestimation method is that despite this uncertainty in the data (i.e.,outlier), it simply isolates the influence of this particular

0

10

20

30

40

50

60

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Err

ors (

m)

Samples

Absolute Errors (GPS2008 - Prediction2008)

Robust Model

Linear Regression

Artificial Neural Network

Fig. 4. Deviations of the predicted 2008 shoreline positions from the actual measuredGPS position in 2008.

observation from the rest of the sample used in the estimation. Thisway, it avoids the trap of falls rejection and falls retention (seeHampel et al. 1986).

For the artificial neural network, the choice of the architecture ofthe training method and the neurons in the hidden layers areimportant factors influencing the response of the predictive model.In this regard, the results of artificial neural network in Table 3 andFig. 4 show significant variations when the neurons in the hiddenlayer are modified. For example, performing a test with 2 and 10neurons, maximum deviations of 126.04 m and 242.08 m wereobtained, respectively, and with 4 neurons in the hidden layers, theresult was 16.4 m.

The values listed in Table 4 were calculated using equations(19)e(23), and include the minimum and maximum values for thedata. The results are (arithmetic mean, standard deviation, RMS, R2

and MAPE) 6.45 m, 10.10 m, 11.90 m, 98.44% and 0.61% for linearregression; 1.56 m, 1.83 m, 2.39 m, 99.94% and 0.14% for the robustestimation; and 3.69 m, 4.03 m, 5.36 m, 99.68% and 0.33% for theneural networks.

The linear regression results were more than 8 m in comparisonto the control shoreline position in 10 test cases, where themaximum deviation was 57.29 m. The best statistic result waspresented by the robust estimation model with the largest devia-tion from the control shoreline being 7.63 m. The artificial neuralnetwork results had a mean value of 3.69 m with the best coeffi-cient of determinant R2 value but a maximum deviation of 16.46 m.

4. Conclusion

In this work, historical photogrammetric data describing thepositional variation of the shoreline in themunicipality of Matinhosin Brazil was combined with GPS-based observations to comparethe predictive capability of three models; linear regression, robustestimation, and artificial neural network. This was achieved bycomparing the short-term model predicted shoreline position forthe year 2008 to the GPS measured position for the same period. Inso doing, temporal and prediction analysis of shoreline was used togenerate digital maps of shoreline and critical erosion sites, thusproviding scientific data, thereby integrating coastal spatial infor-mation that have been used to support decision making processes.The results indicate the importance of continuous monitoring, thusthe Federal University of Paraná must keep monitoring the area.One of the difficulties that deserve a special mention, however, isthe recovery of historical data series. The preparation phase of suchdata in practice takes a great deal of effort.

On the one hand, the result of this comparison demonstratedthe efficiency of the robust estimation model, with residuals of lessthan 8 m. The results of the linear regression model in some caseswere unexpected and completely misrepresent reality, showingresiduals of more than 10 m in seven cases. The artificial neuralnetwork provides an attractive alternative model for prediction tothe robust estimationmodel with residuals less than 16.5 m. On theother hand, this study demonstrated that, despite the uncertainties

R.M. Goncalves et al. / Ocean & Coastal Management 69 (2012) 102e110110

in the data, as often the case for shoreline predictions; the robustestimation method offers the possibility of appropriately dealingwith such uncertainties. Its attractive feature is that users do notneed to delete any portion of the data that appears to be outliers.Robust procedures are, therefore, capable of the following;

1 providing a method of managing outliers in shoreline predic-tion data, and

2 providing a diagnosis and control tool that may be used tocheck the efficiency of traditional shoreline predictionmethods, such as the linear regression.

Acknowledgments

The authors are grateful to the three anonymous reviewers andthe Editor-in-Chief Prof. Victor N de Jonge for their comments,which helped to improve the quality of this manuscript. RMG andCPK acknowledge the support of PROBRAL (CAPES/DAAD) andCNPq; while JLA acknowledges the financial support of the Alex-ander von Humboldt (Ludwig Leichhardt Memorial Fellowship)and Curtin Research Fellowship. RMG, JLA, and CPK would also liketo thank their host Prof. Bernhard Heck and his team at theGeodetic Institute, Karlsruhe Institute of Technology, for providinga conducive working atmosphere. The authors thank K. Fleming ofGfZ for assistance with the English but take full responsibility. Thisis a TiGER Publication No. 422.

References

Addo, K.A., Walkden, M., Mills, J.P., 2008. Detection, measurement and prediction ofshoreline recession in Accra, Ghana. ISPRS Journal of Photogrammetry &Remote Sensing 63, 543e558.

Aduol, F.W.O., 1994. Robust geodetic parameter estimation through iterativeweighting. Survey Review 32 (252), 359e367.

Álvarez-Ellacuría, A., Orfila, L., Gómez-Pujol, G., Simarro, N. Obregon, 2011. Decou-pling spatial and temporal patterns in short-term beach shoreline response towave climate. Geomorphology 128 (3e4), 199e208.

Arbib, M.A., 2003. The Handbook of Brain Theory and Neural Networks. Massa-chusetts Institute of Technology.

Awange, J.L., Aduol, F.W.O.,1999. An evaluation of some robust estimation techniquesin the estimation of geodetic parameters. Survey Review 35 (273), 146e162.

Awange, J.L., 2012. Environmental Monitoring Using GNSS. Springer,Berlin-New York.

Boak, E.H., Turner, I.L., 2005. Shoreline definition and detection: a Review. Journal ofCoastal Research 21 (4), 688e703.

Crowell, M., Letherman, S.P., Buckley, M.K., 1991. Historical shoreline change: erroranalysis and mapping accuracy. Journal of Coastal Research 7 (3), 839e852.

Crowell, M., Douglas, B.C., Leatherman, S.P., 1997. On forecasting future U.S. shore-line positions: a test of algorithms. Journal of Coastal Research 13 (4), 1245e1255.

Demarest, J.M., Leatherman, S.P., 1985. Mainland influence on coastal transgression:Delmarva Peninsula. Marine Geology 63, 19e33.

Douglas, B.C., Crowell, M., Leatherman, S.P., 1998. Considerations for shorelineposition prediction. Journal of Coastal Research 14 (3), 1025e1033.

Douglas, B.C., Crowell, M., 2000. Long-term shoreline position prediction and errorpropagation. Journal of Coastal Research 16 (1), 145e152.

Fenster, M.S., Dolan, R., Elder, J.F., 1993. New method for predicting shorelinepositions from historical data. Journal of Coastal Research 9 (1), 147e171.

Fenster, M.S., Dolan, R., Morton, R.A., 2000. Coastal storms and shoreline change:signal or noise? Journal of Coastal Research 17 (3), 714e720.

Galgano, F.A., Douglas, B.C., Leatherman, S.P., 1998. Trends and variability ofshoreline position. Journal of Coastal Research 26, 282e291.

Galgano, F.A., Douglas, B.C., 2000. Shoreline position prediction: methods anderrors. Environmental Geosciences 7 (1), 1e10.

Goncalves, R.M. Short-term trend modeling of the shoreline through geodetic datausing linear regression, robust estimation and artificial neural networks, 2010.Ph.D. thesis. Geodetic Sciences Post-graduate Program, Federal University ofParana (UFPR), Curitiba, Brazil, p. 152.

Gorman, L., Morang, A., Larson, R., 1998. Monitoring the coastal environment; partIV: mapping, shoreline changes, and bathymetric analysis. Journal of CoastalResearch 14 (1), 61e92.

Hagan, M.T., Menhaj, M.B., 1994. Training feedforward networks with the Mar-quardt algorithm. IEE Transactions on Neural Networks 5 (6), 989e993.

Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A., 1986. Robust Statistics:the Approach Based on Influence Functions. Wiley, New York.

Hayati, M., Shirvany, Y., 2007. Artificial neural network approach for short term loadforecasting for Illam region. World Academy of Science. Engineering andTechnology 28, 280e284.

Haykin, S., 1999. Neural Networks e A Comprehensive Foundation. McMasterUniversity Hamilton, Ontario, Canada, Pearson Education.

Hecky, R.E., Newbury, R.W., Bodaly, R.A., Patalas, K., Rosenberg, D.M., 1984.Environmental impact prediction and assessment: the southern Indian lakeexperience. Canadian Journal of Fisheries and Aquatic Sciences 41 (4),720e732.

Huber, P.J., 1964. Robust estimation of a location parameter. Annals of MahtematicalStatistics 35, 73e101.

Huber, P.J., 1981. Robust Statistics. John Wiley & Sons, New York.Kröse, B., Smagt, P.V.D., 1996. An Introduction to Neural Networks, eighth ed. The

University of Amsterddam.Krueger, C.P., Goncalves, R.M., Heck, B., 2009. Surveys at the coast of Parana, Brazil,

to determinate the temporal coastal changes. Journal of Coastal Research 1,632e635.

Lera, G., Pinzolas, M., 2002. Neighborhood based Levenberg-Marquardt algorithmfor neural network training. IEE Transactions on Neural Networks 13 (5), 1200e1203.

Li, R., Di, K., Ma, R., 2001. A comparative study of shoreline mapping techniques. In:The 4th International Symposium on Computer Mapping and GIS for CoastalZone Management, Nova Scotia.

Li, R., Keong, C.W., Ramcharan, E., Kjerfve, B., Willis, D., 1998. A costal GIS forshoreline monitoring and management e case study in Malaysia. Surveying andLand Information Systems 58 (3), 157e166.

Metropolitan Borough of Sefton, 2002. Shoreline monitoring annual report 2001/2002. http://www.sefton.gov.uk/pdf/TS_cdef_monitor_20012.pdf (accessed14.11.08.).

Mitishita, E.A., Kirchner F.F. Digital mono-diferential restitution of airphotos appliedto planimetric mapping. In: International Archieves of Photogrammetry andRemote Sensing, vol. XXXIII, Part B4. Amsterdam 2000, pp. 655e662.

Schaible, B., Lee, Y.C., 1996. Fuzzy logic models with improved accuracy andcontinuous differentiability. IEEE Transactions on Components, Packaging, andManufacturing Technology 19 (1), 37e47.

Soares, C.R., Vobel, I., Paranhos Filho, A.C. 1995. The marine erosion problem inMatinhos municipality. In: Land Ocean Interactions on the Coastal Zone, 1995,São Paulo, pp. 48e50.

Souza, C.R.G., 2009. Coastal erosion and the coastal zone management challenges inBrazil. Journal of Integrated Coastal Zone Management 9 (1), 17e37.

Thieler, E.R., Pilkey Jr., O.H., Young, R.S., Bush, D.M., Chai, F., 2000. Behavior forU.S. coastal engineering: a critical review. Journal of Coastal Research 16 (1),48e70.

Tzatzanis, M., Wrbka, T., Sauberer, N., 2003. Landscape and vegetation responses tohuman impact in sandy coasts of Western Crete, Greece. Journal for NatureConservation 11 (3), 187e195.

Veloso-Gomes, A., Barroco, A.R., Pereira, C.S., Reis, H., Calado, J.G., Ferreira, M.D.C.,Freitas, M. Biscoito, 2008. Basis for a national strategy for integrated coastalzone management e in Portugal. Jornal of Coast Conservatin 12, 3e9.