A comparison between three short-term shoreline prediction models

Ocean & Coastal Management 69 (2012) 102e110

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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 a

Department of Cartography Engineering, Federal University of Pernambuco (UFPE), Geodetic Science and Technology of Geoinformation Post Graduation Program, Recife 50670-901, PE, Brazil b Department of Spatial Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, Australia c Geodetic Science Post Graduation Program, Federal University of Parana (UFPR), Box 19.001, 81.531-990 Curitiba, PR, Brazil d Geodetic Institute, Karlsruhe Institute of Technology, Engler-Strasse 7, D-76131 Karlsruhe, Germany e 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

a b s t r a c t

Article history: Available online 23 August 2012

Monitoring and management of shorelines along populated coastal areas is a very important task, but remains a difficult endeavor. The historical information used for short-term analysis and prediction are always underpinned by uncertainties associated with old data. Predictions of shoreline positions normally 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 necessary to obtain related cartographic information, which included temporal shoreline data obtained from orthophotos. In this contribution, photogrammetric together with GPS data are used to compare the capability of three shoreline prediction models; linear regression, robust parameter estimation, and neural network to predict the 2008 Parana shoreline position, which is then validated using the GPS measured position of 2008. The results indicate a MAPE (Mean Absolute Percentage Error) of 0.61% for the linear regression, 0.14% for the robust estimation, and 0.33% for the artificial neural network method. Although the coefficient of determinant (R2) value for the neural network was the best, i.e., 0.997 compared to 0.994 for the robust model and 0.984 for the linear regression, its maximum deviation from the control values (i.e., 16.46) was almost twice that of robust model (7.63). On the one hand, the robust estimation model provides a more suitable approach for managing outliers in shoreline prediction, and also validating traditional methods such as linear regression. On the other hand, the neural network method offers an alternative approach to the robust prediction model. The results of the study highlightthe importance of a model choice for predicting the shoreline position. Ó 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 severe erosion problems (e.g., Souza, 2009). Beaches are sources of revenue for many countries through tourism, and as such, necessitate proper management in coastal areas. For the efficient management of beaches, however, one of the important tasks remains that of accurately monitoring the coastal shorelines. Gorman et al. (1998) presented an overview of the methods of data collection for coastal monitoring and baseline studies using * Corresponding author. E-mail addresses: [email protected] (R.M. curtin.edu.au (J.L. Awange), [email protected] (C.P. (B. Heck), [email protected] (L.dosS. Coelho).

Goncalves), J.Awange@ Krueger), [email protected]

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

satellite images, aerial photography, profile surveys and bathymetric records. Remote sensing is important for creating base maps able to interpret landform changes and quantify shoreline movement. The hydrographic and also large-scale topographic maps are considered the primary sources of information for shoreline positions and volumetric change computations. The use of profile surveys can also contribute to evaluating shoreline changes and computing beach volume changes along and across the shore. The importance of coastal monitoring information, e.g., on the state of the current erosion and the possible prediction of its impact have been shown by Li et al. (1998) to be critical to decision-making in coastal management. Li et al. (1998) point to the fact that coastal monitoring information could be useful in land-use zoning, construction setbacks, and relocation implementations. They give a practical example of a digital topographical map that is overlaid with a cadastral map and an erosion condition map, which is used to

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find lots and parcels that are affected by coastal erosion. Another area that requires shoreline monitoring is that of the integrated coastal zone management (ICZM), where such information assist in the process of coastal management and territorial planning/ decision-making by providing essential knowledge about coastal processes and their dynamic evolution (e.g., Veloso-Gomes et al., 2008). ICZM has been shown to be important for example in the detection of landscape and vegetation changes, and to establish the relations between human impact and vegetation in natural coastal dune systems (see, e.g., Tzatzanis et al., 2003). Monitoring of beach stability, therefore, is vital for environmental as well as resource management, and is essential for improving databases of information on shoreline evolution in an area. The Metropolitan Borough of Sefton (2002) listed the benefits of shoreline evolution information as; providing input to shoreline review plans, planned maintenance of coastal defenses, achievement of high government level targets, determination of appropriate design criteria for coastal works, biodiversity action plans, implementation of habitats directive, and leisure and amenity management of shoreline areas. While shoreline monitoring, as discussed above is essential, predicting its future position is equally vital to support coastal management and impact assessment programs, e.g., predicting the necessary future building setbacks from the shoreline to serve as protection for a time comparable to the expected lifetime of new coastal structures, usually 30 or 60 years (Hecky et al., 1984; Crowell et al., 1997). Crowell et al. (1997) point out, however, that determining adequate setbacks require estimating long-term shoreline change trends from historical data. For historical dataset, metric quality and aerial photographs are very important and are some of the most reliable data sets available for coastal zones. Crowell et al. (1997) and Douglas et al. (1998) reckon that often the data used in such predictions are at times temporally poorly sampled historical databases. They point out that short records are not good predictors of shoreline location. Fenster et al. (1993) developed a predictive method that detects short-term changes in the long-term trend and identifies linear or high-order polynomial models that best fit the data according to the minimum description length (MDL) criterion. In this method, only linear models are extrapolated. Predictions shaped or influenced by higher-order polynomial schemes can sometimes be superior to those obtained from linear regressions, but they can also be extremely inaccurate (Crowell et al., 1997). Douglas et al. (1998) stressed the need to incorporate long-term erosion trends and historical record of storms, including their impacts on shoreline positions and beach recovery in predictive models. Thieler et al. (2000) tested the use of mathematical models to predict beach behavior for U.S. and cited lack of hind-sighting and objective evaluation of beach behavior predictions for engineering projects, and incorrect use of model calibration as some of the motivation behind their work. Shorelines are known not to be stable and vary over time. Shortterm changes occur over decadal time scales, or less, and are related to daily, monthly, and seasonal variations in tides, currents, wave climate, episodic events and anthropogenic factors (e.g., Demarest and Leatherman, 1985; Galgano et al., 1998; Galgano and Douglas, 2000). Fenster et al. (2000) describe shoreline movement as a complex phenomenon and outline the difficulties involved in distinguishing long-term shoreline movement (signal) from shortterm changes (noise). Even with the presence of noise in the data, the use of modern accurate surveying techniques has been shown to improve the quality of forecasts. For example, Douglas and Crowell (2000) demonstrated that meaningful deduction was achievable even if the inherent variability of shoreline position indicators remained at the level of many meters. Other than the surveying technique used

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for observation, a challenge for detecting, monitoring, and predicting shoreline movement is to develop an effective mathematical tool capable of identifying and mapping a feature set as “shoreline” according to the data sources available (Boak and Turner, 2005). For instance, exploiting the relationship between shoreline and sea level changes (i.e., using series of sparsely sampled sea-level values as surrogate data for shoreline change), Douglas et al. (1998) developed an algorithm that evaluates some of the predictive methods such as the end point method, linear regression, and minimum description length criterion and established that linear regression gave superior results. The need for an effective mathematical tool is further emphasized by Li et al. (2001) who stress the need to integrate specific coastal engineering and modeling software packages in a GIS environment since they are usually not provided by commercial GIS software system. The problem with the linear regression models is that they work well when the underlying linearity and normal distribution assumptions are fulfilled. However, in some cases, where the data maybe of poor quality, the linear regression assumptions may be violated. Uncertainties in the extracted shoreline data need to be appropriately addressed if they are to be used to predict future shoreline positions to support sustainable coastal management (see, e.g., Addo et al., 2008). Robust estimation models have been proposed 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 from photogrammetric and GPS surveys for the years 2001, 2002, 2005 and 2008, this study discusses the possibility of developing a shortterm prediction model using different sources of temporal geodetic data, with most of them having different accuracies. To achieve this, a comparison and assessment of three different shoreline prediction models; linear regression, robust parameter estimation, and artificial neural network is undertaken. The robust estimation model used in this work has been successfully applied by Awange and Aduol (1999) to estimate geodetic parameters in the case of contaminated observations. The artificial neural network method tested in this paper is termed Neural Network Multilayer Perception (MLP), and was implemented 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 the study 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 been collecting data using GPS surveys over the Parana State coast (see, e.g. Krueger et al., 2009). The data are composed of digital cartographic documents with reports showing the post-processing and accuracy reached in a specific survey. With a proposal to study shoreline changes along the Parana coast, some analog images/ photographs related to the years 1954, 1963, 1980, 1991 and 1997 were retrieved and used for shoreline extraction along a 6 km coastal zone of the Matinhos beaches in the state of Parana, Brazil. 2.2. Study area The study area is located in the Matinhos District along the coast of Parana State, Brazil (Fig. 1), where the beaches (105 km long) form the second smallest littoral along the Brazilian coast, located

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

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 being composed of Pre-Cambrian units of crystalline complexes of igneous and metamorphic rocks. Pleistocene and Holocene sandy sediments of marine origin from a terrace (beach ridges) are found between the shoreline and the foot of the Serra do Mar Mountains. The beaches of Matinhos are subject to both oceanic conditions and ebb tidal delta influence from the neighboring Guaratuba Bay (micro-tidal system). The district of Matinhos, with an area of 117 km2, comprised 29,172 habitants in 2010. Land use is mostly for recreational purposes and in summer, the city becomes more densely populated due to the influx of tourists. The settlement in Matinhos is largely near the shore, where settlements are characterized by constructions at backshore positions or over the beach leading to the destruction of dunes and wetlands, thereby forcing rivers to change course. This was a result of lack of urban planning and the fact that the morphology and coastal dynamic environment was not considered during the settlements. Fig. 2 shows the situation in 2007 and 2008 at the

study area, indicating the prevalence of the problems of coastal erosion. 2.3. Data source Monitoring of shoreline positions nowadays benefit from the state of the art mapping techniques such as global navigation satellite 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 and Ranging). For instance, Mitishita and Kirchner (2000) extracted shoreline position from temporal vertical aerial photographs by the monorestitution technique, which required altimetry information and was dependent on the quality of the photos, the distribution and number of control points, and photo-interpretation for shoreline extraction. In the present study, aerial photographs and GPS observations collected for a 6 km part of the coastal zone were used, where the photogrammetric data was converted to digital orthophotos using

Fig. 2. State of the Matinhos beach in 2007 and 2008 (Goncalves, 2010).

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control points and digital terrain model (DTM). The shoreline positions were then obtained using the monorestitution technique based on photo interpretation. The GPS data was collected during a series of geodetic surveys of the shoreline in Matinhos using the kinematic relative positioning method (see, e.g., Awange, 2012). The GPS survey was undertaken by LAGEH (Laboratory of Space Geodesy and Hydrography) in UFPR (Federal University of Parana) using dual frequency GPS receivers. The base station was installed at Pedra located at 25
490 5.779900 S; 48
310 49.136400 W in Matinhos. All temporal vectors of shorelines from photogrammetry for the years 1954, 1963, 1980, 1991 and 1997 (Fig. 3a) and from GPS for the years 2001, 2002, 2005 and 2008 (see Fig. 3b) were placed in layers and processed in the same geodetic reference frame (World Geodetic System; WGS-84). For distance computations, planar coordinates (UTM) were used. The shoreline positions obtained from photogrammetric data did not always begin and end at the same location, and also had gaps, and as such did not cover all the information at the study site.

2.4. Shoreline prediction models In what follows, brief discussions of the linear regression, robust estimation, and artificial neural network models are presented. 2.4.1. Linear regression model The linear regression model commonly used in shoreline prediction is expressed as

y ¼ ax þ c1;

(1)

105

where x and y are the linear regression vectors, while a and c are the unknown linear regression coefficient and intercept respectively, 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 Ab þ ε;

(2)

with y being an n  1 column vector of observations, A is an n  m design matrix (here m ¼ 2), b is the m  1 vector of unknown linear regression parameters, and ℇ is the n  1 vector of observational errors. Equation (2) can now be expressed in the form of the GausseMarkov estimation model as:


y ¼ Ab þ ε; εw 0; s20 W 1 ;

(3)

whose solution (linear regression) is given by

b b ¼




AT WA

1

AT Wy;

(4)

s 20 ðAT WAÞ1 with the dispersion Df b bg ¼ b and sb 20 ¼ ðεT W ε Þ=ðn  mÞ. Here, bs 20 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 100 m. The temporal positions of shorelines are then obtained from their intersection with the cross-sections. These positions are then used to calculate the shoreline distances xn, related to time tn, which forms the elements of the observation vector y in equation (1). The temporal information, i.e. years, are used in the design matrix A. The linear regression solution (4) holds when there are no gross errors and b b becomes an unbiased estimator of b since the

Fig. 3. Temporal resolution of the photogrammetric and GPS data.

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

y ¼ Efyg þ 2 þ 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:

b b ¼

1
1
AT Wy þ AT WA AT W dy; AT WA

(6)

with the expectation


1
1   E b b ¼ AT WA AT Wy þ AT WA A T W 2 ¼ b þ g ¼ b0 ; (7) and y ¼ Efyg; dy ¼ 2 þ ε: It becomes evident then that b b is a biased estimator of b with the termðAT WAÞ1 AT W 2. Setting g ¼ db, equation (7) leads to

b ¼ b0 þ db:

(8)

2.4.2. Robust estimation model The robust estimation model is defined as follows: Consider x1,x2,.,xn as being randomly independent variables assumed to have a normal distribution described by the probability density function F0(x). If a fraction k(0 < k < 1) of the observations are contaminated by gross errors, then the new distribution will be represented by Huber (1964):

FðxÞ ¼ ð1  kÞF0 ðxÞ þ kHðxÞ;

(9)

where H(x) is the unknown contaminating distribution. The robust estimation procedure estimates parameters from this setup in such a way that the influences of the gross errors in the final estimated parameters are significantly reduced. In adopting a robust estimation procedure, we seek a rigorous adjustment method that will solve equation (6) and give the results that are as close as possible to those of equation (4). One advantage of using robust estimation methods is that they do not immediately delete outlying observations from a given set, rather, they isolate their effects through, e.g., down-weighting. The iterative weighting approach proposed by Aduol (1994), for instance, provides a means of down-weighting bad observations. In this approach, the weights of observations are considered as functions of both the observational variances and the observational residuals. In equation (3), the weight matrix is defined as

W ¼ S1 ;

(10)

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

W ¼




1 R2 þ S ;

(11)

where

2

n1

60 R ¼ 6 4 : 0

0

n2 : 0

: : : :

3 0 07 7; : 5

v ¼ y  Ab b:

The values of v obtained from equation (13) are re-introduced into equation (11) to obtain a new weight matrix W. The procedure for the estimation of b b is then repeated until a sufficient level of convergence is attained. In this iterative approach, the starting initial values of the weight matrix W are those obtained from the linear regression solution in equation (4). This method makes use of the fact that any gross error in the observation will manifest itself in the estimated residuals, which will tend to be larger compared to the others due to the effect of the gross errors. In the re-estimation of the elements of the weight matrix W, a relatively large residual will result in a relatively low weight for the respective observation. The procedure therefore down-weights the influence of outlying observations and in so doing, reduces the effect of bad observations on the estimated quantities. 2.4.3. Neural network In beach morphodynamics applications, Álvarez-Ellacuría et al. (2011) have shown using an example that artificial neural networks can be used to obtain shoreline positions from daily raw video images where the spatial and temporal variability of the beach are split by Empirical Orhthogonal Function (EOF) decomposition. For the setup and development of the artificial neural network (NN), the characteristics of neurons, topology, and training rules must be specified. The adaptation of the initial weights and learning of their behaviour are specified by the rules of training. The algorithms of training of a NN have the feature of adjusting iteratively the weights of connections between neurons until the pair of inputs and expected outputs are obtained, and consequently the relations of cause and effect established. When the setting of a particular problem presented to NNs changes, and the performance model does not fit the situation anymore, it is possible to train the NN with new conditions of input and output to improve the performance (Kröse and Smagt, 1996; Haykin, 1999). The schematic model of an artificial neuron considered as the processor of a neural network was proposed by McCullochePitts, where three basic elements can be identified (see, e.g. Kröse and Smagt, 1996; Haykin, 1999; Arbib, 2003). The first element is a set of synapses or connecting links, each characterized by a weight wkj. The second element relates to a sum of the input signals, weighted by the synapses of the neuron (linear combination) as

vk

m X

wkj $xj :

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

(14)

j¼0

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

yk ¼ 4ðnk Þ:

(15)

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

(12)

nn

(13)

f ðnÞ ¼

1 1 þ eðanÞ

(16)

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

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

f ð nÞ ¼ a

eðbnÞ  eðbnÞ eðbnÞ þ eðbnÞ

(17)

The linear activation function is defined by Equation (18) where a 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 architecture, or the way in which neurons in a network may be arranged, which is an important parameter that restricts the type of problem that can be treated in a specific network. Networks with a single layer of nodes, for example, can only solve linearly separable problems. Recurrent networks, in turn, are more appropriate to solve problems that involve temporal processing (Haykin, 1999). The Multilayer Feed-Forward Network was used with the LevenbergeMarquardt (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 output neurons. There are several types of methods for training networks that are grouped into two categories: supervised learning and unsupervised learning. The case of supervised learning or learning with a teacher has a set of inputeoutput examples. However in the case of learning without a teacher (unsupervised learning), there is neither teacher to oversee the learning process nor are there labeled examples of the function to be learned by the network (see, e.g., Haykin, 1999). The most popular algorithm of error management is the back propagation, were the idea is to correct the errors according to a learning rule. The learning consists of two steps through the different layers of the net according to their direction; one step forward, the propagation, and one step backward, the back propagation (e.g. Kröse and Smagt, 1996; Haykin, 1999; Arbib, 2003). In the forward step, one vector of input is applied to the sensorial nodes and it in effect propagates through the net, layer by layer. This results in one set of output to be produced with a real answer or estimate by the net. During the propagation through the net, all weights are fixed. During the backward step, all weights are adjusted according to a rule of correction of errors (see, e.g. Kröse and Smagt, 1996; Haykin, 1999; Arbib, 2003). Specifically, the answer of the net is subtracted from that desired to produce the error signal. This signal is propagated back through the net, hence the name back propagation. The weights are adjusted to make the answer of the real net closer to the desired output (see, e.g. Kröse and Smagt, 1996; Haykin, 1999; Arbib, 2003). In practice, the use of the back propagation algorithm tends to converge very slowly, requiring a great computational effort. To solve this problem, several techniques have been incorporated to improve the performance in order to reduce the convergence time and the computational effort required. Such techniques include, e.g., the LevenbergeMarquardt algorithm (see, e.g. Hagan and Menhaj, 1994; Lera and Pinzolas, 2002). A test using neural networks is done with the characteristic of non-supervised training. The inputs are the vectors of observations y described in equation (2). The network learns the relationships provided by the input information and finds out what the output for the forward step representing the 2008 predictions are. The data for the year 2008 does not participate in the training. Similarly to the linear regression and robust estimation methods, the 2008 data serves only as a control to verify the answers found by the respective predictive model. It is emphasized that any kind of weight is selected as the input data for the neural network case.

107

Table 1 show the setting used to perform the neural network. It is important to report that we tried many more neural network configurations with different input and hidden layers (or neurons) to check the performance of each model, with the best one selected and compared to the other methods. 2.5. Evaluation criteria Three experiments were performed; the first with the linear regression model (equation (4)), the second with the robust estimation model (equations (10)e(13)), and the third with the neural network (Table 1). To assess the efficiency of the three methods, we computed the following: (1) The mean of the deviation from the GPS 2008 derived shoreline, which we used as a control. This is computed using

x ¼

n 1X ðGPS 2008i  Prediction 2008i Þ; n i¼1

(19)

where (GPS 2008i  Prediction 2008i) denotes the distance between the predicted shoreline and the control shoreline at the cross-section, while i and n are the number of sections. (2) The standard deviation is then obtained from the square root of the average variance of the estimated shoreline position, i.e.,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u 1 X s¼ t ðGPS 2008i  Prediction 2008i  xÞ2 n  1 i¼1

(20)

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

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u 1 X RMS ¼ t ðGPS 2008i  Prediction 2008i Þ2 : n  1 i¼1

(21)

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

Pn ðGPS 2008i  Predictioni Þ2 R2 ¼ 1  Pn i ¼ 1 2 i ¼ 1 ðGPS 2008i  Mean GPS 2008Þ

(22)

(5) Finally, we determine the mean absolute percentage error (MAPE), which measures the accuracy of a fitted time series and usually expresses accuracy as a percentage (see, e.g., Hayati and Shirvany, 2007). Table 1 Artificial neural network settings. Architecture e Multilayer Feed-Forward Networks e Structure Artificial neural network Training method Number of hidden neuron Number of hidden layer Activation function used in hidden layer Activation function used in output layer

MLP LevenbergeMarquardt(LM) 1 4 Hyperbolic tangent Linear

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Table 2 Cross-sections characteristics.

MAPE ¼

Cross-sections Temporal data belonging to the set 10 15 10 7 9 1 5 Total ¼ 57

1963 1963 1954 1954 1954 1963 1963

2001 1980 1963 1963 1963 1980 1980

2002 2001 1980 1980 1980 1997 1997

2005 2002 2001 1991 1991 2001 2001

2005 2002 2001 1997 2002 2002

Degrees of freedom

2 3 2005 4 2002 2005 5 2001 2002 2005 6 2005 4 3

 n   100 X GPS 2008i  Prediction 2008i   GPS 2008i n i ¼ 1

(23)

2.6. The 2008 predictions All historical data are placed in layers and once this is done, one is chosen as a shoreline reference upon which the cross-sections are drawn. In this case, the first line obtained by GPS (2001) was selected as a reference, since it comprised the limits of extension of

Table 3 Results of the prediction models for the 2008 shoreline position compared to that of the GPS survey in 2008 (GPS 2008-model results). Sample number

Number of iteration for robust best solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

12 13 28 1 14 3 1 20 2 2 27 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 21 4 4 3 3 2 2 2 5 7 19 3 1 3 1 3 1 5 1 1 1 3 2 2 2 1 2 1 3 1 7

Robust (GPS2008  Prediction)

Linear regression (GPS2008  Prediction)

Artificial neural network (GPS2008  Prediction)

Absolute value (m)

Absolute value (m)

Absolute value (m)

0.85 1.36 1.01 1.31 0.9 0.93 0.29 0.47 2.13 0.45 2.24 0.68 0.81 1.58 0.23 1.76 0.23 1.61 0.79 1.60 0.78 1.24 0.32 1.29 1.24 0.06 0.50 0.45 0.70 0.12 0.69 1.18 3.01 2.15 0.30 6.00 0.36 0.36 1.57 0.07 4.21 1.46 4.77 0.01 6.73 6.21 0.73 0.28 1.97 0.28 0.68 7.63 2.36 0.27 0.04 1.20 6.20

14.35 14.54 8.94 1.31 5.29 6.48 0.29 7.58 2.20 4.75 12.84 6.54 6.62 3.97 5.23 4.04 5.35 1.61 0.79 1.60 0.78 1.24 0.32 1.29 1.24 3.12 1.85 3.08 3.28 0.36 1.44 3.70 8.57 9.22 28.74 57.29 47.36 13.68 1.57 5.37 4.21 4.51 4.77 1.99 6.73 6.21 0.73 3.23 2.10 1.36 3.28 7.63 2.62 0.27 1.77 1.20 7.00

3.54 3.89 2.06 2.33 1.48 1.35 0.19 0.03 4.62 3.30 4.94 0.34 0.31 0.88 1.05 0.68 1.41 0.73 1.29 0.12 0.79 1.04 0.91 1.96 1.78 7.86 3.98 3.84 2.66 4.28 5.38 0.54 2.61 0.25 2.62 12.09 14.15 16.46 2.32 1.88 2.83 2.30 4.96 1.26 11.81 11.96 3.43 1.18 3.14 2.35 3.05 13.57 12.51 4.35 2.45 1.52 3.33

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the coastline studied. From it, cross-sections with a distance of 100 m were drawn. In practice there are 57 transverse lines and Table 2 presents the respective information; number of cross-sections, temporal information of the data, and the degrees of freedom relative to the linear mathematical model, i.e., the difference between variables and the number of independent equations. For the weight matrix in equation (10), the photogrammetry data are assigned a standard error of 16 m, i.e. 2s following Crowell et al. (1991) who suggests an error of about s ¼ 8 m. GPS measuring accuracy of 2 cm is adopted for the kinematic relative GPS data (see, e.g., Awange, 2012). With this information, the vector of the unknowns comprising the linear regression parameters a and c is estimated. The established linear regression models are then used to predict the shoreline position for the year 2008. This procedure is repeated with the robust estimation and the neural network methods. The results of the predicted shoreline for 2008 from all the three methods under study are compared with that of the 2008 GPS 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 used as the control for the set of 57 samples used in the study. The second column of Table 3 shows the number of iterations performed to obtain the best solution of the robust estimation. When the number of iterations is 1, the solution found by the robust estimation method is essentially the same as that of the linear regression. As can be seen from the results of Table 3, Robust estimation method has the smallest values of the absolute errors, i.e., GPS position of 2008 e the predicted model positions. This is visible in Fig. 4, which presents the deviation of the predicted 2008 shoreline position from the GPS control data of 2008 constructed from 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 the largest absolute error as clearly depicted by the results of the Robust estimation method. This sample could possibly be an outlier due to the uncertainty in its data. The advantage of the Robust estimation method is that despite this uncertainty in the data (i.e., outlier), it simply isolates the influence of this particular

Absolute Errors (GPS2008 - Prediction2008) 60

Robust Model

50

Linear Regression Artificial Neural Network

Errors (m)

40 30 20 10 0

1

5

9

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

Samples Fig. 4. Deviations of the predicted 2008 shoreline positions from the actual measured GPS position in 2008.

109

Table 4 Statistical data. Statistics results

Robust model (m)

Linear regression (m)

Neural network (m)

Maximum Minimum Mean Standard deviation RMS 2 R MAPE

7.63 0.01 1.56 1.83

57.29 0.27 6.45 10.1

16.46 0.03 3.69 4.03

2.39 0.994 0.14

11.9 0.984 0.61

5.36 0.997 0.33

observation from the rest of the sample used in the estimation. This way, it avoids the trap of falls rejection and falls retention (see Hampel et al. 1986). For the artificial neural network, the choice of the architecture of the training method and the neurons in the hidden layers are important factors influencing the response of the predictive model. In this regard, the results of artificial neural network in Table 3 and Fig. 4 show significant variations when the neurons in the hidden layer are modified. For example, performing a test with 2 and 10 neurons, maximum deviations of 126.04 m and 242.08 m were obtained, respectively, and with 4 neurons in the hidden layers, the result 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 the data. 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 linear regression; 1.56 m, 1.83 m, 2.39 m, 99.94% and 0.14% for the robust estimation; and 3.69 m, 4.03 m, 5.36 m, 99.68% and 0.33% for the neural networks. The linear regression results were more than 8 m in comparison to the control shoreline position in 10 test cases, where the maximum deviation was 57.29 m. The best statistic result was presented by the robust estimation model with the largest deviation from the control shoreline being 7.63 m. The artificial neural network results had a mean value of 3.69 m with the best coefficient of determinant R2 value but a maximum deviation of 16.46 m. 4. Conclusion In this work, historical photogrammetric data describing the positional variation of the shoreline in the municipality of Matinhos in Brazil was combined with GPS-based observations to compare the predictive capability of three models; linear regression, robust estimation, and artificial neural network. This was achieved by comparing the short-term model predicted shoreline position for the year 2008 to the GPS measured position for the same period. In so doing, temporal and prediction analysis of shoreline was used to generate digital maps of shoreline and critical erosion sites, thus providing scientific data, thereby integrating coastal spatial information that have been used to support decision making processes. The results indicate the importance of continuous monitoring, thus the Federal University of Paraná must keep monitoring the area. One of the difficulties that deserve a special mention, however, is the recovery of historical data series. The preparation phase of such data in practice takes a great deal of effort. On the one hand, the result of this comparison demonstrated the efficiency of the robust estimation model, with residuals of less than 8 m. The results of the linear regression model in some cases were unexpected and completely misrepresent reality, showing residuals of more than 10 m in seven cases. The artificial neural network provides an attractive alternative model for prediction to the robust estimation model with residuals less than 16.5 m. On the other hand, this study demonstrated that, despite the uncertainties

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in the data, as often the case for shoreline predictions; the robust estimation method offers the possibility of appropriately dealing with such uncertainties. Its attractive feature is that users do not need 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 prediction data, and 2 providing a diagnosis and control tool that may be used to check the efficiency of traditional shoreline prediction methods, such as the linear regression. Acknowledgments The authors are grateful to the three anonymous reviewers and the Editor-in-Chief Prof. Victor N de Jonge for their comments, which helped to improve the quality of this manuscript. RMG and CPK acknowledge the support of PROBRAL (CAPES/DAAD) and CNPq; while JLA acknowledges the financial support of the Alexander von Humboldt (Ludwig Leichhardt Memorial Fellowship) and Curtin Research Fellowship. RMG, JLA, and CPK would also like to thank their host Prof. Bernhard Heck and his team at the Geodetic Institute, Karlsruhe Institute of Technology, for providing a conducive working atmosphere. The authors thank K. Fleming of GfZ for assistance with the English but take full responsibility. This is a TiGER Publication No. 422. References Addo, K.A., Walkden, M., Mills, J.P., 2008. Detection, measurement and prediction of shoreline recession in Accra, Ghana. ISPRS Journal of Photogrammetry & Remote Sensing 63, 543e558. Aduol, F.W.O., 1994. Robust geodetic parameter estimation through iterative weighting. Survey Review 32 (252), 359e367. Álvarez-Ellacuría, A., Orfila, L., Gómez-Pujol, G., Simarro, N. Obregon, 2011. Decoupling spatial and temporal patterns in short-term beach shoreline response to wave climate. Geomorphology 128 (3e4), 199e208. Arbib, M.A., 2003. The Handbook of Brain Theory and Neural Networks. Massachusetts Institute of Technology. Awange, J.L., Aduol, F.W.O., 1999. An evaluation of some robust estimation techniques in 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 of Coastal Research 21 (4), 688e703. Crowell, M., Letherman, S.P., Buckley, M.K., 1991. Historical shoreline change: error analysis and mapping accuracy. Journal of Coastal Research 7 (3), 839e852. Crowell, M., Douglas, B.C., Leatherman, S.P., 1997. On forecasting future U.S. shoreline positions: a test of algorithms. Journal of Coastal Research 13 (4), 1245e 1255. 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 shoreline position prediction. Journal of Coastal Research 14 (3), 1025e1033. Douglas, B.C., Crowell, M., 2000. Long-term shoreline position prediction and error propagation. Journal of Coastal Research 16 (1), 145e152. Fenster, M.S., Dolan, R., Elder, J.F., 1993. New method for predicting shoreline positions from historical data. Journal of Coastal Research 9 (1), 147e171.

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