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Ocean & Coastal Management 85 (2013) 77e89

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Ocean & Coastal Management

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Assessment of spatial and temporal variations of high water markindicators

Xin Liu a,*, Jianhong (Cecilia) Xia a, Michael Kuhn a, Graeme Wright b, Lesley Arnold c

aDepartment of Spatial Sciences, Curtin University, Bentley, WA 6845, AustraliabResearch and Development, Curtin University, Bentley, WA 6845, AustraliacGeospatial Frameworks, Perth, WA 6000, Australia

a r t i c l e i n f o

Article history:Available online 9 October 2013

* Corresponding author. Tel.: þ61 449920138 (mobE-mail addresses: xin.liu2@postgrad.curtin.edu.au

(J.(Cecilia) Xia), M.Kuhn@curtin.edu.au (M. Kuh(G. Wright), Arnolds@iinet.net.au (L. Arnold).

0964-5691/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.ocecoaman.2013.09.009

a b s t r a c t

The high water mark (HWM) is commonly used as a boundary for coastal management and planning.Due to the dynamic nature of the coastal environment, the determination of HWM can be difficult andmay vary based on the indicators unique to the location.

Using remote-sensing image analysis techniques, this study evaluates the spatial and temporal vari-ation of HWM based on several indicators. These include vegetation lines, frontal dune toe, mean highwater spring (MHWS)/mean higher high water (MHHW), and high water lines (HWL). Other linearboundaries defined by agencies for various applications are also used as indicators.

For improved coastal property management, this study also uses an enhanced Spatial Continuity of theSwash Probability (SCSP) model as a HWM indicator by excluding the runup parameter regarding theSpatial Continuity of Tide Probability (SCTP). In order to better account for sudden shape changes, theextended instead of the simple Hausdorff distance has been used to measure the seasonal variation ofHWM position. Monte Carlo simulation of DEM data and Fractal Dimension (FD) techniques were used toexamine spatial uncertainties due to both the precision of input data and the processing techniques used.

Two case study areas in Western Australia with varying coastal conditions have been selected toevaluate the approach. These are Coogee Beach in South Fremantle and Cooke Point in Port Hedland.Results for both study areas indicate that spatial variations of HWM due to seasonal changes are aboutone order of magnitude larger than variations due to uncertainties in the input data. This behaviour,while present at both study areas, is more significant at Coogee Beach having a sandy beach with highwave energy.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

All High Water Mark (HWM) indicators, such as berm crest orvegetation line, are located within the swash zone. The swash zoneis the transition zone between the subaqueous (below water) andsubaerial (above water line) zones of a beach, and is intermittentlycovered and exposed by wave action (Hughes et al., 2010). Theswash zone is of particular interest to coastal engineers, plannersand researchers, as it defines the transition between ocean and dryland. It is also important for coastal property management, whichrequires knowledge of spatial and temporal variations of HWM dueto higher cross-shore sediment transportation rates in the swashzone (Hughes et al., 2010; Masselink and Russell, 2006). Alsina and

ile).(X. Liu), C.Xia@curtin.edu.aun), g.l.wright@curtin.edu.au

All rights reserved.

Cáceres (2011) also indicated that the beach close to the shoreline isa highly dynamic areawhere the transportation of coastal sedimenttakes place frequently. However, the underlying dynamic mecha-nism remains poorly understood (Masselink and Russell, 2006).

The aim of this study is to evaluate the spatial and temporalvariation of the selected indicators using image analysis techniques.To assess the seasonal variation of each indicator’s position, theextended Hausdorff distance is applied as a metric indicator tobetter account for sudden shape changes. Factors influencing theprecision of HWM determination including random errors and thecomplexity, due to both the accuracy of input data and the pro-cessing techniques used, were evaluated by the methods of MonteCarlo simulation and fractal dimension (FD) calculation as twoparts of spatial variation for HWM indicators. Two case study arease Coogee Beach in South Fremantle and Cooke Point in Port Hed-land, with distinctly different coastal types were selected toexamine the methods. The Airborne Light Detection And Ranging(LiDAR) topographic data for the coast at Port Hedland was of poor

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e8978

quality and hindered further analysis. This was improved using theKriging interpolation method.

This paper includes a background section providing an overviewon HWM indicators in use for coastal management and planning,HWM variation in space and time, and its uncertainty due to theprecision of input data (Section 2). This is followed by an expla-nation of the study areas, materials, data, and methods used in thisstudy (Section 3). All results and discussion regarding spatial andtemporal variations of HWM for the selected study areas are pro-vided in Section 4. A brief discussion and conclusion are presentedin Section 5 including limitations of the present study and futureresearch required.

2. Background

2.1. HWM indicators

The most common HWM indicators are mean high water spring(MHWS) and mean higher high water (MHHW). MHWS is used forcoasts with a predominately semidiurnal tidal feature, and meanhigher high water (MHHW) is used for coasts that have mostly adiurnal tidal regime. These HWM indicators are easily applied whenthere is sufficient tidal information available. In the case of insuffi-cient tidal information, land surveyors prefer to use field evidence toestablish the position of HWM when defining the boundary linebetween private and public ownership (Morton and Speed, 1998).

Normally, the area between the vegetation line and the bermcrest is defined as the beach (Bauer and Allen, 1995; Rooney andFletcher, 2000). Therefore, most features lying on the beach canbe considered as an HWM indicator (Pajak and Leatherman, 2002).This is because physically thesemarkings indicate the runup of highwater and thus the HWM line (Hicks et al., 1989; Simon, 1993;Williams-Wynn, 2011).

Previous studies considered several common types of HWMindicators in addition to tidal records. These include the boundarybetween dry and wet sand referred to as the high water line (HWL)(Moore et al., 2006), dune toe (Williams-Wynn, 2011), and theseaward limit of vegetation (Williams-Wynn, 2011) referred to asthe vegetation line. These shoreline features are good indicators ofwater level and are often used as boundary indicators between landand water (Coutts, 1989; Gay, 1965; Maiti and Bhattacharya, 2009;Moore, 2000; Morton and Speed, 1998). The process to identifythese indicators has been discussed by Liu et al. (2012), using imageanalysis techniques.

In addition to HWM indicators, government authorities providetheir own interpretation/determination of HWM. For example, this isthe case in Western Australia where the state’s surveying and titleinformation authority, Landgate, and the state’s Department ofTransport (DoT) use different HWMs side-by-side. The heights sug-gested by DoT and Landgate were obtained from long-term obser-vation of the shoreline features and tide and wave effects on thecoastal areabyexperienced surveyors. To integrate the landandwaterinto a holistic system, Liu et al. (2012) developed a method based onthe spatial continuity of the swash probability (SCSP) to determinethe position of HWM for the purpose of coastal hazard planning un-der extreme situations. In this study, the SCSP is enhanced bychanging the model parameters, the spatial continuity of tide prob-ability (SCTP), which is specifically defined for coastal propertymanagement purposes. Historically, tidal datum-based HWM in-dicators have their roots in the development of common law forproperty management, and the origin of using tide levels to defineHWM can be traced back to the sixteenth century (Cole, 1997). Theprinciple of determining SCTP is essentially the same as the methoddefined by Liu et al. (2012), but without taking into account the effectof wave runup.

HWM indicators, depending on their usage and theway they aremeasured, can be divided into two components: vertical and hor-izontal HWM indicators. Tidal datum based HWM indicatorsbelong to the vertical HWM category, while naturally occurringrepresentations, such as vegetation line, belong to the horizontalHWM category. HWM is not always tied to a specific high waterheight; sometimes a cadastral boundary that represents HWM ismapped as a horizontal line. Moreover, a vegetation line, dune toeor HWL left on the beach have been adopted as natural represen-tations of a horizontal HWM.

Although HWMmay exist in two different forms (horizontal andvertical), the aim of HWM determination coincides with delin-eating the coastal boundary. When HWM is defined vertically, it ispossible to derive its horizontal HWM position on the coastal zoneor in the imagery with spatial information. This means that, tosome extent, a transformation can be used to derive one form ofHWM from another.

However, HWM indicators are highly variable over time andtend to be at the same location for short periods only. They are alsonot available on every beach. Analysis of tide gauge records and thebeach profile variations show that there is temporal variation ofshorelines over different time scales. These variations range fromshort-term daily changes to seasonal changes and multi-decadalchanges (Pugh, 1996).

2.2. Monitoring beach profile variations

A significant number of quantitative analysis studies have beenconducted on the seasonal changes in beach profiles (Aubrey, 1979;Aubrey and Ross, 1985; Shepard, 1950; Weishar and Wood, 1983;Winant et al., 1975). These studies show that seasonal morphologychanges cause the most dominant temporal variations (Masselinkand Pattiaratchi, 2001); with large changes of the beach profileoften occurring between summer and winter. Changes betweenspring and autumn beach profiles are often almost identical (Larsonand Kraus, 1994). However, the results of these temporal variationstudies are limited as they are only based on two-dimensional pa-rameters (e.g., transection lines), which may result in large data gapsbetween established profiles (Hapke and Richmond, 2000).

The most traditional way to examine changes in beachmorphology is through field surveys (Austin and Masselink, 2006;da Fontoura Kle and de Menezes, 2001; Eliot et al., 2006), and thetools to study change have improved over time. Today, the mostcommonly used equipment for field surveys is based on Real-TimeKinematic Global Positioning Systems (RTK-GPS) with centimeteraccuracy (Dail et al., 2000; Travers, 2009). However, field surveysare very time consuming and labour intensive and cannot coverlarge areas in a short time frame.

Alternatively, video techniques can be used to monitor theevolution of beach morphology (Lippmann and Holman, 1989,1990). However, video is costly in terms of setup and mainte-nance of the video camera. In addition, video techniques are highlysensitive to field conditions, such as lighting conditions.

Airborne Light Detecting And Ranging (LiDAR) techniques canbe used to capture onshore features over large areas in a short timeframe (Armaroli et al., 2004; Zeng et al., 2001) using DEMs, thus it isa complementary technique used to fill data gaps between groundprofiles. Moreover, when applied in a repetitive manner, LiDARenables three-dimensional analysis of temporal beach profile var-iations. More potential applications of LiDAR-derived DEMs forstudying the coastal morphology has been illustrated in studies byHapke and Richmond (2000) and Mitasova et al. (2002). However,it is important to note that this technique can also result in datagaps. These are caused by obstructions on the ground, such asvegetation and built-up areas.

Fig. 1. Map of the study area at South Fremantle.

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e89 79

Inherent in all the monitoring techniques identified above is theoccurrence of large data gaps (Bater and Coops, 2009; Hodgson andBresnahan, 2004; Hodgson et al., 2005). Therefore, spatial inter-polation methods are needed to fill in these data gaps because theinformation existing in the data gaps may still be useful for furtheranalysis, especially for the DEM.

The Kriging method can be used to successfully interpolate datagaps in ground profiles. Compared to other interpolation methods,such as nearest neighbour, spatial averaging, and inverse distanceweighting, Kriging (Bailey and Gatrell, 1995; Griffith, 1988) has beenproven to providemore accurate (minimumvariance) and less biasedestimates when applied to beach morphology (Oliver and Webster,1990; Wong et al., 2004). However, Kriging may rely on thesmoothness assumption of the interpolated surface (Li and Heap,2008) and it is restricted to modelling the first and second order ef-fects in spatial analysis (Emery, 2006). Nonetheless, Li and Heap(2008) state that when observations are insufficient to computevariograms, the gap in sparse data can be satisfactorily interpolatedusing the Kriging method.

Another technique for quantifying the variation of spatial ob-jects over time is to use image analysis principles. Most recently,time series analysis has been developed to detect temporal changesof beach morphology using a series of remotely sensed images. The(periodic) time series is usually analysed in terms of general trends,seasonal variations and residual components (Herold, 2011; Jacquinet al., 2010; Jong et al., 2011). One of the models developed toillustrate this idea is the “Breaks For Additive Seasonal and Trend”(BFAST) (Verbesselt et al., 2011, 2012). However, the application ofthis model depends highly on whether a sufficient number of im-ages are available as input data (e.g. more than two per year whenanalysing seasonal variations and annual variation). When imagedata is limited (e.g. only two images), a simplified method can beused to identify temporal variations by comparing the two differentimages (Andrews et al., 2002). While this is a relatively simplemethod the results may be highly biased due to seasonal andlonger-term variations.

Studying the spatial variation of complex but linear objects, suchas HWM, requires the quantification of their spatial relationship (e.g.distance). Various methods are available to measure the spatialdistance between two linear objects, such as minimum Euclideandistance (Peuquet,1992) and surface “in between” (McMaster, 1986).However, neither of these methods is able to determine the truemathematical distance (Hangouët, 1995). In this regard, the Haus-dorff distance was introduced as a “safe and systematic” distance tocalculate the largest minimum distance (see Section 3.5.1) betweentwo vector polylines (Hangouët, 1995). However, the Hausdorff dis-tance can be rather unstable when there is a local sudden change inthe shape of the measured objects (Min et al., 2007). Therefore, anextended Hausdorff distance was introduced by Min et al. (2007) byremoving (smoothing) sudden changes before calculation of theHausdorff distance. This is considered as a more accurate measure.This paper adapts the extended Hausdorff distance method to assessspatial distances between HWM lines captured in summer andwinter seasons based on different indicators.

2.3. Spatial variation of the shoreline feature from thedetermination process

The variation in a shoreline feature from the determinationprocess can be expressed as spatial precision, thus indicatingapparent spatial variations. When using imagery based on remotesensing techniques, variations in extracted shoreline features canbe expressed by their disagreement between the feature on ageographically registered map and the corresponding position onthe Earth’s surface.

Spatial variations of shoreline features may arise from datasource inaccuracies and interpolation errors (Ruggiero and List,2009). The errors from interpolation processes can occur in thepre-process stages of the determination process, such as identifi-cation of the shoreline position on aerial photography, and post-process stages when it is represented on the map to display thedata (Shi, 2009). When image analysis is applied in shorelinefeature position determination, pre-processed spatial variations aremainly determined by the classification accuracy and image regis-tration; while if the shoreline position, including HWM, is calcu-lated from a statistical model, the variations are mostly due to theaccuracy of the model itself.

During post-process, HWM in Australia is usually determined asa height above the AHD. When HWM is positioned on the ground,as is the horizontal cadastre, any inaccuracies (including errors anduncertainties) of the digital elevation model (DEM) used to extractbeach profiles based on the AHD can lead to spatial variations thatcan affect the position of derived results such as the beach slopeand contour lines (Hunter and Goodchild, 1997; Oksanen andSarjakoski, 2005).

Traditionally, these errors and uncertainties are categorised intothree groups: 1) gross errors or “blunders”, 2) systematic errors and3) random errors (Cooper, 1998; Wise, 2000). Due to their magni-tude, “blunders” are easily detected and removed prior to dataprocessing. These errors are often associatedwith faulty equipmentand errors in the data collection process (Wechsler and Kroll, 2006).Systematic errors follow a consistent pattern and are often inherentin the procedures used to generate the DEM (Fisher and Tate, 2006),are normally characterised by the RootMean Squared Error (RMSE).

In contrast to systematic errors, random errors can only bequantified through repeated experiments. Currently, due to thecomplexity of processing algorithms, it is not well understood howrandom errors are introduced and propagated through a DEM(Wechsler and Kroll, 2006). Therefore, the uncertainty present inderived products from the DEM cannot be easily determined.

The classical approach to examine the uncertainty is through errorpropagation. This can be done in two ways: 1) developing analyticalerror models, and 2) constructing simulation models (Zhang andGoodchild, 2002). As indicted by Fisher and Tate (2006), analyticalerrormodels are a relatively simpleway to represent the uncertaintyof a land feature from imagery when the analytical model is simple.Error simulation models can be used if the analytical model is com-plex. These models rely on stochastic simulation, and can be divided

Fig. 2. Map of the study area at Port Hedland.

Fig. 3. Original LiDAR points captured at Port Hedland 1995.

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e8980

into two groups: unconditioned and conditioned models. Byconsidering the observations at the same sample location, the con-ditional simulation model with a geostatistical method, which takesinto account the spatial autocorrelation of the simulated features, ismore widely used (Fisher, 1998; Holmes et al., 2000; Kyriakidis et al.,1999). In the stochastic simulation process, previous studies mainlyfocus on the Gaussian error model combined with the Monte Carlomethod, especially when analysing DEM uncertainties (Davis andKeller, 1997; Holmes et al., 2000; Oksanen and Sarjakoski, 2005;Wechsler and Kroll, 2006). However, their application in the evalua-tion of random errors in HWM indicators has not been thoroughlystudied. Therefore, understanding random errors in determination ofHWM is one of the focuses of this study.

Furthermore, due to topographic data complexity, the deter-mination of HWM indicators is not straightforward, and this mayalso influence spatial variation in HWM indicators (Thompsonet al., 2001). A way to quantify the spatial complexity of HWMlines is using the fractal dimension (FD). The FD method has beenextensively applied to understand the complexity of spatial pat-terns and their variation (Burrough, 2006; Palmer,1988). The fractalconcept was first introduced by Mandelbrot (1967) in order toillustrate irregular patterns that cannot be analysed by traditionalEuclidean geometry. While Euclidean geometry only allows di-mensions with an integer number, the important concept of the FDis that it also allows for non-integer dimensions. The FD increasesas the complexity of the spatial pattern increases. This has beenapplied in studies that measure plant development (Corbit andGarbary, 1995) and variation (Palmer, 1988), characterisingcomplexity in earthquake slip and identifying the regressiveecological succession (Alados et al., 2003). However, one of themost commonly used examples to illustrate the fractal concept is tomeasure the length of a coastline (Jiang and Plotnick, 1998;Mandelbrot, 1967; Phillips, 1986; Schwimmer, 2008). Because allHWM indicators can be considered as different representations of

Table 1Beach information on the two study areas (Gozzard, 2011).

Near shore Fore shore Back shore proxi

CoogeeBeach

Sand and seagrass meadows

Low tideterrace

Foredune e stablto prograding

CookePoint

Rock pavement Rockplatform/tempestite/segmented beach

Low calcarenite c

the coastline, the FD was considered suitable to capture the topo-graphic complexity of HWM indicators.

3. Study areas, materials and methods

3.1. Study areas

Two study areas in Western Australia were chosen to evaluatethe spatial and temporal variation of HWM indicators developed inthis study. These are Coogee Beach in South Fremantle and CookePoint in Port Hedland (Figs. 1 and 2). Both sites have distinctiveshoreline characteristics and were chosen as being representativeof typical shore conditions and therefore pertinent for testing themethods being developed. Table 1 summarises the basic shorelinefeatures of these study areas (Gozzard, 2011).

Coogee Beach in South Fremantle is a wave-dominated reflec-tive straight beach with a diurnal tidal feature and a micro tidalrange of 0.7 m (Short, 2004). The site chosen extends approxi-mately 500 m along the coast and is 70m in cross-shore width. Thewave breaking types in South Fremantle are either plunging(81.06%) or spilling (18.94%). This is estimated from the long-termwave information statistical records recorded every hour by thewave rider buoy in Cottesloe (approximately 12 km north of theSouth Fremantle study area) (Tremarfon Pty Ltd., 2011).

In contrast, Cookie Point at Port Hedland is a tidal-dominated sandflat headlandwith a semi-diurnal tidal feature and amacro tidal rangeof 6m(Short, 2004). This site extendsapproximately2300malong thecoast and 200m in cross-shorewidth. Thewave breaking types, basedon an analysis of ten years of historic wave data, are either plunging(92.07%) or spilling (7.93%). The significant wave height (Hs) data andwave peak periods (Tp)were recorded every hour by Beacon 16 in PortHedland (19 km from the study area) (Tremarfon Pty Ltd., 2011).

mal Back shore distal Geology substrate Coastal exposure

e Prograded barrier Unclassified Low

liff Transgressivedune barrier

Calcarenite Moderate/high

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e89 81

3.2. Data used and outline of the methods

The imagery used to capture both study areas and extract HWMindicators was supplied by Landgate and captured using a LeicaADS80 Digital Camera. The South Fremantle regionwas captured inFebruary 2010 with a ground resolution of 0.1 m (Landgate, 2010).The DEM for the same area, also reflecting the summer beachmorphology, was derived from LiDAR data captured in February2008 and produced by WA Department of Water (Department ofWater, 2008). The DEM has a vertical accuracy of 0.15 m and hor-izontal accuracy of 0.6 m. Another DEM (with a vertical accuracy of0.3 m and horizontal accuracy of 0.5 m), reflecting winter beachmorphology, was generated by Landgate (2012) from digital aerialphotography captured in August 2011.

Imagery for the Port Hedland study area was captured inNovember 2009 at a ground resolution of 0.2 m (Landgate, 2009b).The DEM, representing the summer beach morphology, wascreated by Landgate (2009a) from LiDAR data with a vertical andhorizontal accuracy of 0.2 m and 1.0 m respectively. As the pointdensity of the LiDAR data captured at South Fremantle (for bothseasons) and Port Hedland (November 2009) is very high, the pointcloud was resampled to a grid using an Inverse Distance Weightingalgorithm. This is considered sufficient to provide a high qualityDEM. However, the LiDAR points captured to represent the beachmorphology at Port Hedland in winter time (July in 1995) were notdense enough to provide a high quality DEM (Fig. 3). Therefore,further interpolation is required (Landgate, 1995).

The methodology applied in this study is outlined in Fig. 4. TheHWM indicators (see top box in Fig. 4 and Section 2.1) are to be

Fig. 4. Methodology framework.

evaluated from both a spatial and temporal variation viewpoint.This evaluation determines the precision and the stability of theHWM indicators. As the seasonal variation of the beachmorphology is the dominant temporal variation, the position ofeach HWM indicator in winter and summer were compared, andthen measured using the extended Hausdorff distance. The spatialvariation from the determination process (both pre-process andpost-process), which evaluates the precision of HWM indicators,was assessed using Monte Carlo simulation and FD to determinerandom errors and the complexity of the HWM indicators,respectively. In addition to DEM accuracy, there are two otherfactors that may influence the precision of the determined HWM e

image classification accuracy and model accuracy. These were theparameters used to calculate the HWM indicators by Liu et al.(2012).

3.3. Kriging

The first step of the evaluation method on spatial and temporalvariation of HWM indicators is to interpolate an increased densityof data points using the Kriging method (see Section 2.2). Kriging isused to predict the values of the missing DEM data by the sum ofthe surrounding weighted values of the observed DEM data (Oliverand Webster, 1990):

bZðs0Þ ¼XNi¼1

liZðsiÞ (1)

in which, Z(si) is the observed value at location i, li is the weight atthe location i, s0 is the location to be estimated and N is the numberof observed data used to predict bZðs0Þ.

The weights li were estimated based on spatial autocorrelationtheory using semivariogram models as follows (Burrough, 2001;Oliver and Webster, 1990):

gðhÞ ¼ 12VarfFðxiÞ � FðxiþhÞg (2)

where g(h) is the estimated variance between two observed datapoints, h is the distance between two observed data points and F(x)semivariogram function.

3.4. Spatial variation due to errors resulting from the HWMdetermination processeprecision

Asmentioned in Section 2, errors or uncertainty arising from thedata pre-process or post-process, contribute to the variation inextracted HWM indicators and in turn may influence the location.Errors arising from the pre-process, including the shoreline featureclassification and swash height modelling (wave runup model)accuracy, have been studied by Liu et al. (2012). However, therandom errors of the indicators derived from the DEM data and thecomplexity of the HWM indicators, which contribute to the post-process errors, are still unexamined.

In this paper, the influence of random error is studied usingMonte Carlo simulation and spatial autocorrelationmethods.When

Table 2Semivariogram models used for Kriging interpolation.

Model Mean error (m) Root-Mean-Square (m)

Rational Quadratic �0.815 � 10�3 0.151Gaussian �0.100 � 10�2 0.159Circular �0.242 � 10�3 0.153Spherical �0.223 � 10�3 0.153Stable �0.360 � 10�3 0.151

Fig. 5. Semivariogram model representing the autocorrelation of the original LiDAR data at Port Hedland.

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e8982

the HWM indicators are drawn on the map, spatial variation of thelocation of an HWM indicator may occur due to the quality/accu-racy of the DEM and the complexity of the indicator itself. Sys-tematic errors of the HWM determination process associated withthe accuracy of the DEM can be directly understood and evaluated,while the effects of random errors on positioning HWM indicatorsand the process of drawing lines on maps have not previously beensufficiently understood. This problem is explored in this paper.

3.5. Conditional simulation of DEM values

The Monte Carlo simulation method is based on the assumptionthat only random errors are present and that they are normallydistributed (Gaussian distribution) with a constant mean value andstandard deviation. In this study, for each pixel i, the mean and thestandard deviation are derived from the simulated DEM data andare expressed respectively as:

U�pSIMij

�¼

Xmj¼1

pSIMij =m (3)

sU

�pSIMij

�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXmj¼1

�pSIMij � U

�pSIMij

��2=m� 1

vuut (4)

in which m is the total number of simulations (here m ¼ 100),and pSIMij represents the simulated DEM value for pixel i at time j. Toget more realistic random simulation results, the following condi-tion is implicit: that the simulated elevation values, on average, areequal to the elevation values of the original DEM with the standarddeviation equal to the given accuracy (RMSE) of the DEM.

Furthermore, the pSIMij values are usually dependent on

neighbouring values, therefore the elevation simulation modelalso includes spatial dependency described by the spatial auto-correlation of the simulated values (Hunter and Goodchild, 1997).A number of methods have been developed to model such spatialdependencies (Wechsler and Kroll, 2006), including

neighbourhood autocorrelation, mean spatial dependence, andweight spatial dependence. In this study, weight spatial depen-dence, was chosen because it incorporates the notion of spatialautocorrelation, and implemented in combination with thesemivariogram model.

For the Monte Carlo method 100 DEM simulations were carriedout to achieve stable results (Heuvelink, 2006). The correspondingindicators were re-extracted from the 100 DEM simulates, andcompared with the original DEM pixels intersected with the posi-tion of the HWM indicators by the RMSE. This is one of the mostcommon tools to measure the derived simulation differences fromthe original data (NOAA Coastal Services Center, 2011).

3.5.1. Fractal dimension of HWM indicatorsThe fractal dimension of the HWM indicators is estimated using

the logelog relationship (Theiler, 1990):

LogðLðsÞÞ ¼ ð1� DÞLogðsÞ þ b (5)

where L(s)¼ N,s is the length of the HWM line along a coast, whichequals the length of a spatial unit s multiplied by N, the number ofunits needed to cover the complete HWM line. As the spatial unitdecreases in length, the length of the HWM line increases. D is thefractal dimension and b is the residual. As indicated by Mandelbrot(1982), the value (1 � D) is assigned to the slope of Equation (5),which can be estimated using Least Squares regression of the lengthof the HWM line and the combined length of all spatial units used.

After determining and assessing all the factors that govern theprecision of an extracted HWM line during the determinationprocess, the final spatial variation (precision) was obtained bytotalling the absolute error and multiplying this with the relativeerrors.

3.6. Seasonal variation of HWM position e stability

In this paper, the positions of the lines for each HWM indicatorwere first derived separately from the two DEMs representingsummer and winter beach morphology (Section 3.2). The seasonal

Fig. 6. Cross-validation on the Kriging interpolation.

Fig. 7. Kriging interpolated DEM at Port Hedland.

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e89 83

variation (winter and summer) of the corresponding lines repre-senting the HWM positions was evaluated by measuring spatialdistances between them using the extended Hausdorff distance.

3.6.1. Extended Hausdorff distanceThe Hausdorff distance is a max-min distance, which was first

introduced in image analysis by Huttenlocher (1993). Subsequently,Hangouët (1995) applied this method in the study of spatial vari-ation of vector features in GIS. Given two finite point setsA ¼ fa1;/; apg and B ¼ fb1;/; bqg, defining a vector feature, theHausdorff distance is defined as:

HðA;BÞ ¼ maxfhðA;BÞ; hðB;AÞg (6)

where

hðA;BÞ ¼ suppa˛A

(infpb˛B

kpa � pbk)

(7)

and

Table 3Spatial variation of HWM based on different indicators from determination process (precision) at South Fremantle study area.

HWM indicators Classification accuracy Model accuracy (m) DSM error (m) Topographic complexity (FD) Precision (m)

Accuracy Random

Pre-process Post-process

HWL 1.0000 N/A 0.1500 0.0020 1.0032 0.1525MHHW N/A N/A 0.1500 0.0020 1.0032 0.1525Landgate N/A N/A 0.1500 0.0020 1.0032 0.1525DoT N/A N/A 0.1500 0.0010 1.0582 0.1598SCTP N/A N/A 0.1500 0.0010 1.0127 0.1529SCSP N/A 0.38 0.1500 0.0090 1.0021 0.5401Dune toe line 1.0000 N/A 0.1500 0.0020 1.0836 0.1647Vegetation line 1.1370 N/A 0.1500 0.0050 1.1840 0.2087Average 0.2105

Table 4Spatial variation of HWM based on different indicators from determination process (precision) at Port Hedland study area.

HWM indicators Classification accuracy Model accuracy (m) DSM error (m) Topographic complexity (FD) Precision (m)

Accuracy Random

Pre-process Post-process

HWL 1.1140 N/A 0.2000 0.1040 1.1010 0.3729MHWS N/A N/A 0.2000 0.0690 1.0860 0.2921Landgate N/A N/A 0.2000 0.0790 1.0860 0.3030DoT N/A N/A 0.2000 0.1140 1.0890 0.3419SCTP N/A N/A 0.2000 0.0300 1.1840 0.2723SCSP N/A 0.3800 0.2000 0.0240 1.1900 0.7188Dune toe line 1.1860 N/A 0.2000 0.0030 1.1670 0.2810Vegetation line 1.4810 N/A 0.2000 0.0520 1.1920 0.4449Average 0.3784

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e8984

hðB;AÞ ¼ suppb˛B

(infpa˛A

kpa � pbk)

(8)

in which pa and pb are the points in the point sets A and B,respectively, while k$k represents “some underlying metric be-tween points of the sets A and B” (Min et al., 2007). In this study, themetric used is the classical Euclidean distance, p refers to thesegment, and A and B are two segment sets. The length of thesegment corresponds to the horizontal accuracy of the DEMs fromwhich the HWM lines were derived. Therefore, the first step tocalculate the Hausdorff distance between line l and n is to divideeach line into segments, then determine the shortest distance fromthe segment of line l to the closest segment of the line n, and choosethe largest value as the distance between the corresponding seg-ments. The same process is applied to calculate the distance fromline n to line l, and the larger of the two is adopted as the Hausdorffdistance between the two lines. However, as indicated in Section2.2, sudden changes of the shape of the lines may significantly in-fluence the calculation, which is very common for coastal bound-aries due to their complex nature and beach morphology. Thus, theextended Hausdorff distance (Min et al., 2007) was applied in thisstudy to mitigate this problem. The extended Hausdorff distance isgiven by Min et al. (2007):

Hf1f2ðA;BÞ ¼ maxnhf1ðA;BÞ;hf2ðB;AÞ

o(9)

where

hf1ðA;BÞ ¼ minfεi : f1 ¼ wððB4SðεiÞÞXAÞ=wðAÞg (10)

and

hf2ðB;AÞ ¼ min�εj : f2 ¼ w

��A4S

�εj��XB

��wðBÞ (11)

with εi and εj indicating the buffer width (Sð$Þ) for line B and line A,respectively; while wð$Þ is a metric function to measure the lengthof a line. For example, wððB4SðεiÞÞXAÞ represents the length of lineA falling inside the dilated region ðB4SðεiÞÞ. For f1 ¼ f2 ¼ 1, theextend Hausdorff distance equals the Hausdorff distance as indi-cated by Equation (6); in contrast, when f1 ¼ f2 ¼ 0, the extendedHausdorff distance measures the shortest distance between thetwo lines. In this study, both f1 and f2 are assigned the value of 0.5,thus the so called median Hausdorff distance is obtained. This wasrecognised as a rather robust measure (Min et al., 2007) and wasapplied as a metric to evaluate the variation of the position of theHWM lines due to the change in beach morphology over time (inthis instance, between summer and winter).

4. Results and discussion

4.1. Interpolated DEM

The first step of the Kriging interpolation is to derive the sem-ivariogram function of the original data. As shown in Table 2, the“stable function” model with the smallest RMSE and mean errorwas chosen to represent the spatial dependency of the DEM sur-face, in which the major range is 401.470 m, and the partial sill is8.803 m (Fig. 5). The cross-validation shows the results of inter-polation of the DEM are accurate (Figs. 6 and 7). For example, pointsrepresenting the predicted and measured values are intensivelydistributed along the diagonal; while most of the error andstandardised error points are close to 0, and with a small standarddeviation. One reason that high accuracy interpolation results were

Fig. 8. Maps of the DEM at South Fremantle (lower-right) and its first eight conditional simulations.

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e89 85

achieved is that Port Hedland has a low variation of gradient on thesand beach where sparse LiDAR points exist and this reduced themagnitude of elevation interpolation errors.

4.2. Spatial variation due to HWM determination process and dataaccuracy

4.2.1. The effects of DEM random errors on derived HWM indicatorsCompared with the effect of DEM accuracy on the HWM in-

dicators, the random error expressed as spatial uncertaintycontributed less to the spatial variation of HWM (Tables 3 and 4). Asindicated in previous studies (Barber and Shortridge, 2005; Vazeand Teng, 2007), uncertainty analysis is not necessary for highquality LiDAR DEM data because of the high accuracy of the data.However, this study shows that uncertainty may lead to spatialvariations in the derived results e both large and small.

The first eight simulated DEMs for both the South Fremantle andPort Headland study areas are illustrated and compared with theoriginal DEM in Figs. 8 and 9. Each map in the figures representsone possibility that the DEM may exist, due to the uncertainties,which is not necessarily the same as the original DEM. As can beseen from the results in Tables 3 and 4, the apparent spatial vari-ation from such uncertainty is not always small, especially at PortHedland. For example, the uncertainty for the DoT position

(0.114m) is as large as 57% of the systematic error. This can occur fortwo reasons: firstly, the accuracy of the DEM at Port Hedland is notas high as at South Fremantle (see Section 3.2), which would in-crease the variation of the simulation for each pixel; and secondly,the coastal land surface at Port Hedland is not as smooth as that atSouth Fremantle. Thus, only when the beach morphology is highlyregular, the uncertainty analysis of HWM or any other shorelinefeatures determination using high quality LiDAR DEM data is notnecessary.

4.2.2. Fractal dimension of HWM linesAs the FD of the HWM line increases (e.g. close to 2), the line

shows less spatial dependence and becomes more unpredictable;whereas a value approaching 1 indicates there exists a direct spatialrelation in the distribution of the HWM line (Palmer, 1988). Tables 3and 4 show the FD of the HWM lines corresponding to the differentindicators selected. Generally, the topographic complexity of HWMindicators at Port Hedland is larger than that at South Fremantle,due to the complex beach morphology. Correspondingly, thiswould limit the accuracy of the surveyed HWM position at PortHedland. From the results obtained it can be seen that the FD of thevegetation line in both study areas was found to be the largest dueto the highly dispersed distribution of the vegetation zone, indi-cating the highest spatial complexity and variability. Although

Fig. 9. Maps of the DEM at Port Hedland (lower-right) and its first eight conditional simulations.

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e8986

widely adopted as the position of HWM, the results show, it maynot be the most suitable indicator due to its high variability. FromFigs. 10 and 11, it can be seen that the FD will increase as theelevation of HWM increases with their position located morelandward. However, the FD of the DoT HWM line in South Fre-mantle was identified higher than the other indicators around it.

Fig. 10. HWM indicators’ FD at South Fremantle.

This does not follow the general trend of the FD. This may be due tothe position of the DoT HWM indicator located around the berm,thus the high variation of the berm elevation causes a larger un-certainty of the position of the HWM line around it. The Landgateand DoT HWM lines, as well as, MHWS in Port Hedland also havesmaller FD than HWM lines derived from the other indicators.

From Tables 3 and 4, It can be concluded that the two mostimportant sources of spatial variation of HWM arise from the

Fig. 11. HWM indicators’ FD at Port Hedland.

Table 5Seasonal variation of HWM based on different indicators (stability) at South Fre-mantle study area.

HWM indicators Stability (Median Hausdorff distance; m)

HWL 5.91MHHW 4.61Landgate 6.50DoT 9.14SCTP 8.07SCSP 3.00Dune toe line 2.16Vegetation line 1.70Average 5.14

Table 6Seasonal variation of HWM based on different indicators (stability) at Port Hedlandstudy area.

HWM indicators Stability (Median Hausdorff distance; m)

HWL 9.08MHWS 1.11Landgate 1.12DoT 3.13SCTP 2.21SCSP 2.17Dune toe line 1.47Vegetation line 3.38Average 2.96

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e89 87

accuracy of the model used to estimate the wave runup heights andDEM error, which made SCSP more variable over space than theother indicators.

Similarly, the high value on the classification accuracy test,topographic complexity (e.g. FD), and even uncertainty in the DEM,make the identification of the vegetation line on the beach difficult.

The variation for all the other indicators is less than the averagelevel, and the MHWS, MHHW, SCTP and dune toe line resulted inhigher levels of precision in the determination at both study areas.

Fig. 12. Zoom in one detailed area for spatial and temporal variation at SouthFremantle.

In general, the variation arising from the determination at PortHedland is larger than that at South Fremantle, due to the highervariation of the coastal morphology and onshore featuredistribution.

4.3. Seasonal variation of HWM indicators’ position

Tables 5 and 6 illustrate the seasonal variation of position of theHWM indicators at the two study areas. Generally, it can be esti-mated from the tables that seasonal variation of HWM position isalmost one order of magnitude larger than the spatial variation dueto errors from the determination process. Furthermore, there is alarger seasonal variation of the position of the HWM indicators atSouth Fremantle than at Port Hedland. Although, the data used toanalyse the variation at Port Hedland have a temporal gap of 14years (1995e2009) between the summer and winter linesevaluated.

From Table 5 and Fig. 12, it can also be estimated that at SouthFremantle the sediment accumulation in the low wave runoffsummer months makes the HWM lines “lower” (moving seaward)than the ones in the high wave runoff winter months. This situationis more apparent on the backshore where indicators of DoT andSCTP lie. In contrast, the occurrence of highly irregular morphologyat the foreshore, due to the high energy swash in the winter season,makes the HWM lines lying in this zone not as “straight” as duringthe summer time. Three indicators, SCSP, dune toe line and vege-tation line, which are close to the vegetation zone, show leasttemporal variation that is mainly due to the low level of the swashprobability.

At Port Hedland (Table 6 and Fig. 13), except for HWL, the hor-izontal offsets of the HWM lines for all other indicators are less than4 m between the summer and winter. Although the rocky coastalzone at Port Hedland may stabilise the HWM lines, the differenteffect of tide and wave activity on the beach morphology andsediment transport is another factor explaining the low variation.This has already been shown by Davis Jr. (1985), who states that

Fig. 13. Zoom in one detailed area for spatial and temporal variation at Port Hedland.

X. Liu et al. / Ocean & Coastal Management 85 (2013) 77e8988

tide, compared with wave breaks, plays a passive and indirect rolein the beach evolution and its change in profile.

According to Masselink and Pattiaratchi (2001), seasonal beachcycles are mainly due to the seasonal variation of the wave energylevel. Compared with the wave-dominated Coogee Beach at SouthFremantle, the wave energy at the tide-dominated Cooke Point atPort Hedland is much lower (Short, 2004). This may partly explainwhy less apparent seasonal variation of HWM occurs at Port Hed-land. HWL shows largest horizontal offset between the two seasonsat both study areas and concurs with the conclusions of Pajak andLeatherman (2002) that the “HWL position can be highly vari-able”. In contrast, the position of dune toe shows small variationconsistently at both study areas.

In sum, the seasonal variation of HWM is more significant atSouth Fremantle, because of the coastal conditions; while thevariation of HWMdue to determination process is more apparent atPort Hedland, where complex beach morphology exists and thedata to represent it is not as good as that at South Fremantle.

5. Conclusions

This paper has evaluated the spatial and temporal variation ofHWM lines derived from different HWM indicators usingremote-sensing image analysis. Although the HWM location canbe observed through physical features, the certainty in its hori-zontal position is very weak. It is in the order of meters, or eventens of meters if the terrain is close to horizontal. As such, theaccuracy of the surveyed position is limited by the knowledge ofthe position of the feature, rather than limited by the surveyinstrumentation and methods used. As a boundary or realproperty, the HWM is understood to be an ambulatory (moving)boundary and is temporal. A survey today does not determine theextent of rights to land and seashore tomorrow. It is also un-derstood that the location of the HWM is not to the order ofaccuracy that other fixed boundaries are located due to the na-ture of the boundary.

Compared with the spatial variation due to the determinationprocess and data accuracy, the study shows that seasonal variationcontributes almost one order of magnitude larger, than the generalvariation of HWM indicators. From the analysis result of fractaldimension, this research also demonstrated that the complex beachmorphology can cause uncertainty of the HWM position on thecoast. In other words, the more regular the coastal land surface is,the less uncertain HWMdeterminationwill be. To visually illustratethe variation of the HWM line corresponding to the different in-dicators for both the spatial and temporal view, variation mapshave been presented and analysed.

Results of this study indicate the dune toe line to be the bestHWM position in terms of the small variation over both time andspace. However, this study only focuses on the position of HWM interms of spatial and the temporal variation, which reflects theprecision and the stability of HWM. Other factors such as proba-bilistic estimates (e.g. the risk of inundation) should also beincluded in the process of establishing HWM. Including this andother decision factors from experts of different disciplines, such ascoastal management and coastal planning, a multi-critical decisionmodel for HWMdetermination is suggested to be established in thefuture. A major challenge of such a model is the establishment of“optimum” weights applied to each decision factor included.Another limitation of this study is that the temporal variationanalysed at Port Hedland may mix with seasonal and annual vari-ations, although the effect of annual variations may not be as sig-nificant as seasonal variations. Whether the annual variation couldbe ignored or not requires further analysis with sufficient timeseries images being available.

Acknowledgements

The authors thank Ric Mahoney, Murray Dolling, and RussellTeede for their interest and for providing a number of suggestionsin this research. Wave data for the two sites was provided byTremarfon Pty Ltd. The images for both sites and the DEMs for PortHedland in two seasons and for South Fremantle in winter timewere captured by Landgate, WA; while the Department of Water,WA, provided the DEM for South Fremantle in summer time. Thisresearch was funded by the Department of Spatial Science, CurtinUniversity, and Landgate, WA. The work was also supported by theCooperative Research Centre for Spatial Information, activities ofwhich are funded by the Australian Commonwealth’s CooperativeResearch Centres Programme. My sincere thanks also go to Asso-ciate professor Jennifer Whittal at University of Cape Town, for theinsights she has provided.

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