a simple equilibrium model for predicting shoreline change
TRANSCRIPT
Coastal Engineering 73 (2013) 191–202
Contents lists available at SciVerse ScienceDirect
Coastal Engineering
j ourna l homepage: www.e lsev ie r .com/ locate /coasta leng
A simple equilibrium model for predicting shoreline change
M.A. Davidson ⁎, K.D. Splinter, I.L. TurnerSchool of Marine Science & Engineering, Plymouth University, Drake Circus, Plymouth, Devon, PL8 4AA, UK
⁎ Corresponding author. Tel.: +44 1752 584740; fax:E-mail address: [email protected] (M.A. D
0378-3839/$ – see front matter. Crown Copyright © 20http://dx.doi.org/10.1016/j.coastaleng.2012.11.002
a b s t r a c t
a r t i c l e i n f oArticle history:Received 10 September 2012Received in revised form 3 November 2012Accepted 7 November 2012Available online 23 December 2012
Keywords:Shoreline modellingEquilibriumHindcastngPredictionHysteresisMorphology
This contribution describes the development, calibration and verification of a 1-D behaviour-orientated shorelineprediction model. The model primarily encapsulates shoreline displacement forced by wave-driven cross-shoresediment transport. Hysteresis effects are shown to be important and are included in the model, whereby pres-ent shoreline change is influenced by past hydro-/morpho-dynamic conditions. The potential magnitude ofshoreline change increaseswith incidentwave power and the degree of disequilibrium. The latter disequilibriumterm (Ωeq−Ω) is expressed in terms of the time-evolving equilibrium (Ωeq) and instantaneous (Ω) dimension-less fall velocities and dictates the direction of shoreline movement. Following Wright et al. (1985) the equilib-rium fall velocity is defined as a function of the weighted antecedent conditions and is a proxy for the evolvingbeach state. The decay rate of the weighting function used to computeΩeq is a model free parameter determinedby calibration against measured data, which physically reflects the degree of observed ‘memory’ of the system.The decay in amplitude of this weighting function with time is controlled by a ‘memory decay’ term (ϕ),where the weighting reaches 10%, 1% and 0.1% at ϕ, 2ϕ and 3ϕ days prior to the current calculation time. Themodel is applied to two multi-year (6+ years) data sets incorporating hourly wave and weekly shoreline mea-surements, from two contrasting energetic sites in SEAustralia. The first is the relatively dissipative, straight GoldCoast (QLD) and the second is a more intermediate embayed beach at Narrabeen (NSW). The model shows sig-nificant skill at hindcasting shoreline change at both sites, predicting approximately 60% of the total shorelinevariability. The Gold Coast shoreline is dominated by a strong seasonal signal. Conversely, at the Narrabeen em-bayment, shoreline variability (and morphology) is more dynamic, responding at storm frequency. Evidencesuggests that there is a strong coherence between the shoreline position and morphodynamic state and thatboth have response times characterised by ϕ. It is hypothesised that optimised ϕ values in the shoreline modelphysically relate to the efficiency of sediment exchanges between the shoreface and offshore bars and the prev-alence of one- or two-dimensional horizontal circulation. The general success of this new shoreline model forhindcasting the observed shoreline behaviour at two distinctly different open-coast sites suggests that this ap-proach may be suitable for broader application.
Crown Copyright © 2012 Published by Elsevier B.V. All rights reserved.
1. Introduction
Shoreline change is affected by amultitude of complexprocesses op-erating at various time- and length-scales (Larson and Kraus, 1995;Miller and Dean, 2004). At shorter time-scales (bdecade) on energeticcoastlines wave-dominated cross-shore transport, longshore transportgradients, wave set-up and storm surge are the dominant processesdriving shoreline change. At longer time-scales eustatic and isostaticsea-level change may become significant along with slower sedimentfluxes to/from the shoreface from Aeolian transport, fluvial sourcesand sub-closure depths. Further factors may include anthropogenicbeach and shoreface replenishment and the impact of complex geology/man-made structures.
Coastal managers, scientists and engineers have long sought arobust and practical methodology for the prediction of shoreline
+44 (0) 1752 586 101.avidson).
12 Published by Elsevier B.V. All rig
change along sandy coastlines, over time-scales spanning severalyears to decades. The existing models, which at the present timecome closest to satisfying these requirements generally include aconsiderable level of empiricism and may be termed top-down ordata-driven models (Avdeev et al., 2009; Cowell et al., 2003;Horrillo-Caraballo and Reeve, 2010; Karunarathna et al., 2009;Różyński, 2003). Probably the best-known and most widely used ex-ample is the GENESIS model (Hanson, 1989; Nam et al., 2011), whichis applicable to predicting generalised planform shoreline evolutionfor the special case where alongshore gradients in sediment trans-port dominate.
Recently there have been several advances in the field of long-termbut relatively high-resolution (weeks to months) shoreline predictiondue to predominantly cross-shore sediment transport processes. Paral-lel but independent work utilising multi-year observational datasetsfrom the southeast coast of Australia (Davidson and Turner, 2009;Davidson et al., 2010) and the west coast of the USA (Yates et al.,2009, 2011) have separately reached similar conclusions. These new
hts reserved.
192 M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
studies fromopposite sides of the Pacific have concluded that a practicalapproach to hindcasting (and potentially forecasting) of multi-yearshoreline variability may be developed from a combined considerationof the evolving disequilibrium state of a beach through time, and therapidly-varying forcing caused by prevailing wave conditions. Thesecore ideas build upon earlier disequilibrium concepts introduced byseveral authors; notable examples include the work of Wright et al.(1985), Plant et al. (1999) and Miller and Dean (2004), where theevolution of beach-state, sand bars and shorelines were examined,respectively.
The seminal work of Wright and Short (1984) on classifyingmorphodynamic beach state has been internationally recognisedand widely adopted. Their classification system identified six keybeach states ranging from a low energy reflective condition, throughfour dynamic and rapidly changing intermediate states, to an ener-getic dissipative extreme. Wright et al. (1985) identified that beachstate is well correlated with the dimensionless fall velocity (Dean,1973; Gourlay, 1968) and that future beach state is sensitive to ante-cedent values of this same parameter.
Davidson et al. (2010) (hereafter DLT10) presented a model thatutilised Wright et al. (1985) disequilibrium concept as a basis forhindcasting and forecasting shoreline change, but neglected anydependence of the equilibrium condition on antecedent conditions.The new work presented here further investigates the extent towhich these concepts can be applied to the prediction of shorelinechange, including consideration of antecedent conditions and provid-ing a conceptual discussion on the extent to which shoreline andmorphological change are linked.
Yates et al. (2009) developed a shoreline model based on theconcept of energy disequilibrium that explicitly included feedback,whereby the predicted shoreline change was functionally dependenton the previous modelled shoreline location and equilibrium energystate. These authors demonstrated that their model was capableof hindcasting shoreline variability with impressive skill at severalUS Pacific coast sites exhibiting similar beach slopes. More recently,Yates et al. (2011) further demonstrated that the calibratedmodel was also transportable to other sites with similar (but notidentical) grain size and incident wave energy without significantloss of skill, increasing confidence that equilibrium models of thisgenre have the potential to be generically applicable, and thatrelationships between model free parameters and measureableenvironmental variables, such as grain size, might yet be an achiev-able goal.
The present study focuses on shoreline change due to primarilycross-shore transport processes; implicitly accounts for wave set-upeffects and simplistically allows for shoreline trends resulting fromlongshore gradients in sediment transport and other processes discussedabove. The simplicity of the present model easily facilitates futureextension to include more explicitly other key processes. In thiscontribution the modelled shoreline is the cross-shore location ofa contour located at (or near) mean sea level, measured relative to
Table 1Environmental variables and descriptions for the Gold Coast and Narrabeen study sites.
Site Mediangrainsize(mm)
Fallvelocity(m/s)
Averageintertidalgradient
WaveBuoylocationrel. to site
WaveBuoydepth(m)
Max/meanwaveheights(m)
Record-averapeak waveperiods (s)
Gold Coast 0.25 ≈0.033 ≈0.02 2 km E 16 7.001.13
9.42
Narrabeen 0.3–0.4 ≈0.055 ≈0.12 10 km SE 85 6.871.62
9.60
an arbitrary datum. The underlying challenge that presently moti-vates the further pursuit of this area of research is to develop a sim-ple and generic modelling tool, which can be used to assess andquantify the likely impacts along sandy coastlines in response to an-ticipated future climatic change.
The model is tested on data collected from two exposed but con-trasting beaches on the Australian East Coast that are described inSection 2. This is followed by an account of the new model develop-ment (Section 3). The model results including an objective assess-ment of model skill and a sensitivity analysis to key parameters arepresented in Section 4. The key findings are discussed in Section 5and the main conclusions are summarised in Section 6.
2. Study sites and test data (shoreline andwave climate time-series)
Two datasets comprising six years of hourly waves and weeklyshoreline observations are used in this present study. Data wereobtained from two contrasting sites, the Gold Coast, Queensland,and Narrabeen, New South Wales (Harley et al., 2011a), both locatedalong the southeast coastline of Australia. Both sites are semi-diurnaland micro-tidal, with spring tide ranges less than 2 m. Although ex-posed to a similar wave and tidal regime there are notable differencesbetween the two study sites (Table 1).
Narrabeen can exhibit the full range of beach states from reflectiveto dissipative; it is typically observed to transition rapidly betweenlongshore-bar-trough (LBT) to low-tide-terrace (LTT) intermediatesurfzone and beachface morphology (Short, 1985; Wright et al.,1985). Morphodynamically, Narrabeen is most frequently an interme-diate, rip-dominated, embayed beach. Two-dimensional horizontalcirculation is common (Short, 1985), influencing the rate of sedimentexchange between the inner bar and shoreface (Splinter et al., 2011),causing recurring bar welding, then detachment, resulting in the ob-served rapid landward/seaward movement of the shoreline. Wright etal. (1985) and Harley et al. (2009) demonstrated a strong hysteresisin the beachmorphology and shoreline location respectively. Specifical-ly,Wright et al. (1985) found that the future morphodynamic state wasstrongly dependent on the antecedent hydro-/morpho-dynamic condi-tions. Similarly, Harley et al. (2009) found that storms followingpersistent high-energy conditions had significantly less impact on theshoreline position when compared to storm events that followed lowenergy conditions. Harley et al. (2009) attributed this observation tothe presence/absence of a dissipative offshore bar and flatter/steeperbeach profile following antecedent storm/calm conditions. EOF analysisby Harley et al. (2011b) suggests that the dominant mode of shorelinevariability (≈60%) within the Narrabeen embayment is attributable toonshore-offshore sediment exchange, setting an appropriate bench-mark for shoreline models based primarily on cross-shore sedimenttransport (as in the present contribution).
By comparison, the morphology of the long, straight Gold Coastbeach is quite different. The beach is morphodynamically moredissipative with less rapid transitions between beach states than
ge Standarddeviation ofShorelineSeries (m)
Comments
9.1–11.2 20 km straight beach exposed to all wave conditions. Persistentdouble bar system. Less dynamic system in terms of beach statetransitions. Strong longshore transport.
7.4–8.0 3.5 km long swash aligned embayment. Wave exposure variesconsiderably around the embayment. Highly dynamic in termsof beach state transitions with frequent exchanges of sedimentbetween the beach-face and inner bar. Relatively frequent barwelding events occur at this site. Strong 2DH circulation.
193M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
those observed at Narrabeen. The Gold Coast exhibits a persistentdouble offshore bar system (Ruessink et al., 2009), strong longshorecurrents and sediment transport (approximately 500,000 m3 yr−1)and a dominantly 1D horizontal circulation. The Gold Coast beachgradient is generally lower than is observed at Narrabeen meaningthat break point bars form further seawards from the shoreline.The combination of these factors means that sediment exchangesbetween offshore sandbars and the beach are much less frequent atthis contrasting site.
Weekly shoreline data were obtained from Argus coastal imagingstations operating at the Gold Coast and Narrabeen (Holman andStanley, 2007; Turner et al., 2006). At both sites, the weekly shorelinepositions used here were obtained by alongshore-averaging of a shore-line contour over a length-scale appropriate to the site (1 km for theGold Coast, 400 m for Narrabeen), to minimise the influence of morelocalised alongshore variability (e.g. beach cusps). The mean sea level(MSL) and the MSL+0.7 m contours were used at the Gold Coast andNarrabeen respectively. Wave data were obtained from local offshorewave-rider buoys at both sites (Table 1).
At both sites three different longshore locations are analysed. Theselocations are shown in Fig. 1. A site-specific coordinate systemwas usedat each site, with a longshore-orientated y-axis. Within these systemsthe location of shoreline measurements used here are centred aty=−1000 m, 900 m, 2000 m (Gold Coast); and y=2200 m, 2600 m,3200 m (Narrabeen). In previously reported work Davidson andTurner (2009) and DLT10 did not consider the Narrabeen site andonly utilised data from y=−1000 m at the Gold Coast, as the othertwo Gold Coast locations are in the lee of (y=2000 m) or immediatelyadjacent to (y=900 m) an artificial surfing reef, and both locations
Fig. 1. Locations of the six shoreline locations studied at the Gold Coast [y=−1000 m, 9
were subject to extensive beach nourishment immediately prior tothe commencement of monitoring (Turner et al., 2006). By includingthese three Gold Coast datasets (where significantly different andindeed opposing shoreline trends under the same offshore forcing areobserved), as well as the three new Narrabeen datasets, a rigorousand robust test of the previous DLT10 and subsequent new model isachieved.
3. Model development
This section begins by defining several objective measures ofmodel skill (Section 3.1) that are subsequently used to assess predic-tive model performance. This is followed by a brief assessment of theDLT10 model on which the present model is developed (Section 3.2).Identified shortcomings of the DLT10 model are used to inform thenew model formulation presented in Section 3.3.
3.1. Model skill assessment
A variety of measures are used here to objectively assess modelskill. The first is a linear squared-correlation (r2) between the mea-sured (x) and modelled (xm) shoreline position. Whilst this methodis useful for exploring correlations between the measurements andpredictions it is possible that the series may be well correlated buthave large residuals. For this reason more rigorous comparativemethods are also used, which compare the model residuals with asuitable baseline (xb). The choice of baseline is somewhat arbitrary.Here, both a linear fit to the shoreline series and the prior DLT10model for shoreline position are used to assess improved model
00 m and 2000 m] (left) and Narrabeen [y=2200 m, 2600 m and 3200 m] (right).
194 M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
performance. The first comparative method, the Brier Skill Score(Sutherland and Soulsby, 2003; van Rijn et al., 2003), also has the ad-vantage of considering measurement error Δx:
BSS ¼ 1−∑ x−xmj j−Δx½ �2∑ x−xbð Þ2 ð1Þ
Positive Brier Skill Scores (BSS) indicate a significant improvementrelative to the base line used and values in excess of 0.0, 0.3, 0.6, 0.8are typically described as ‘poor’, ‘fair’, ‘good’ and ‘excellent’.
Finally, the Akaike's information criterion AIC (Akaike, 1974;Kuriyama, 2012) values were computed, which are specificallydesigned to compare models with a different number of free param-eters (m):
AIC ¼ n log2π þ 1½ � þ n logσ2 þ 2m ð2Þ
Here n is the total number of samples and σ2 is the variance ofthe model or baseline residuals. The difference between the baselineand model AIC (ΔAIC) exceeding 1.0 represents a significant modelimprovement.
3.2. Previous developments
Davidson and Turner (2009) presented a two-dimensional profilemodel that demonstrated considerable skill when applied to thehindcasting of weekly shoreline change spanning a six-year moni-toring period at the Gold Coast in Australia. This model was later sim-plified by DLT10 to provide a one-dimensional shoreline predictionmodel of the form:
dxdt
¼ bþ cΩk Ω0−Ωð Þ ð3Þ
Where dx/dt is the time-varying cross-shore shoreline coordi-nate; Ω=H/wsT is the time-varying dimensionless fall velocity,which in turn is a function of H the offshore significant wave height,ws the sediment fall velocity and T the peak wave period (after Dean,1973; Gourlay, 1968). Ω0 is a constant equilibrium value of the di-mensionless fall velocity and b, c, k are model free parameters.
Previous DLT10 Model C
Sho
relin
e P
ositi
on (
m)
2000 2001 2002 2003160
180
200
220
240
260
Time
Sho
relin
e P
ositi
on (
m)
Previous DLT10 Model C
2004 2005 2006 200720
30
40
50
60
70
80
Fig. 2. Application of the previous DLT10 model to a) the Gold Coast (y=−1000 m) and b)results are much poorer for Narrabeen. The thickness of the data curve (grey) indicates the
Conceptually, the term in the parenthesis, the disequilibrium, dictatesthe potential direction (onshore versus offshore) of sand movement.Shoreline erosion (i.e., a negative/landward movement of the shoreline)occurs when waves are steeper than the equilibrium (Ω0bΩ), with thepotential rate of shoreline response being proportionate to the degreeof disequilibrium (Ω0−Ω) at each moment in time. The time-varyingforcing of shoreline change is also driven by the prevailing wave condi-tions, parameterised by the product of the terms Ωk and a response ratecoefficient c (ms−1). Inclusion of the exponent k permitted a non-linearshoreline response to wave forcing. Finally, the linear trend term b inEq. (1), is independent of forcing, and simplistically accounts for anypotential longer-term shoreline trends that may be present — for exam-ple, a gradient in longshore transport—which is otherwise not accountedfor. The model was shown to be very quickly calibrated using a least-squares method. Notably, DLT10 demonstrated that the above modelshowed significant forecasting skill, over time-scales of several years,whendrivenbyMonte Carlo simulations of syntheticwave data and com-pared with unseen shoreline time series.
Fig. 2 shows the comparisons between the measured shorelineseries and the DLT10 model predictions (Eq. (3)) for both the GoldCoast (y=−1000 m, Fig. 1a) and Narrabeen (y=2600 m, Fig. 1b).As was previously reported the model performance at the GoldCoast is well correlated to the measured data (r2 =0.58, 99% signifi-cance level=0.04) and is a substantial improvement to a simple lin-ear trend model (r2 =0.03, see Table 2). In contrast, although theDLT10 model qualitatively identifies some of the more extreme ero-sive events at Narrabeen, and therefore generally out performs a simplelinear fit (refer to Table 2), the model performance is clearly muchpoorer (r2 =0.38) at this second site, with much of the correlation at-tributed to the trend term (b, r2 =0.14). Fundamentally, themagnitudeof the modelled shoreline variability is greatly under-estimated, andeven the direction of shoreline change is sometimes opposite to that ob-served, leading to the necessary conclusion that an inappropriate equi-librium condition is being applied.
3.3. New model development
The inability of the previousDLT10model to consistently forecast thedirection of shoreline change at the Narrabeen site implies that whilst a
omparison − Gold Coast
2004 2005 2006 2007
DLT10 ModelData: y=−1000m
(Years)
omparison − Narrabeen
2008 2009 2010 2011
DLT10 ModelData: y=2600m
Narrabeen (y=2600 m). Notice that whilst the model performs well at the Gold Coast,potential measurement error.
Table2
Summaryof
mod
elpa
rametersan
dskillsscores
forh
indc
asts(q
uantitieswithno
bracke
ts),calib
ration
{}an
dve
rification
[].H
indc
astsus
eallsurve
ys,calibration
sus
ethefirsth
alfo
fthe
samples
andva
lidations
usethelatter
halfof
thesu
rvey
s.Th
enu
mbe
rof
samples
fortheGoldCo
astan
dNarrabe
enaren=
307,
{153
},[154
]an
dn=
355,
{168
},[168
]resp
ective
ly.A
SR=
Artificial
Surfing
Reef.T
he99
%sign
ificanc
eleve
lfor
r2fortheGoldCo
astan
dNarrabe
enare0.04
,{0.02
}an
d[0.02]
for
hind
cast,calibration
andve
rification
.PositiveBrierSk
illsScores
(BSS
)indicate
asign
ificant
improv
emen
trelativeto
theba
selin
eus
edan
dva
lues
inex
cess
of0.0,
0.3,
0.6,
0.8arede
scribe
das
‘poo
r’,‘fair’,‘goo
d’an
d‘excellent’.Aka
ike'sinform
ation
criterionΔAIC
values
exceed
ing1represen
tasign
ificant
improv
emen
trelativeto
theba
selin
eused
andtake
dueacco
unto
fthe
relative
numbe
rof
free
parametersus
edin
differen
tmod
els.
Location
bc+
c−/c
+ϕ
r2BS
SΔAIC
m/a
×10
−8(m
/s)/
(W/m
)0.5
days
linea
rDLT
10Sh
orefor
Shorefor:lin
earba
selin
eSh
orefor:DLT
10ba
selin
eLine
ar:Sh
orefor
DLT
10:Sh
orefor
GoldCo
ast
y=
−10
00m
1.32
{−6.14
}1.71
5{1.781
}0.51
9{0.458
}>10
00{>
1000
}0.03
0.58
{0.45}
[0.54]
0.60
{0.48}
[0.63]
0.74
{0.65}
[0.61]
0.47
{0.38}
[0.66]
113.5{37.0}
[66.8]
130.4{2.0}[19
.5]
y=
900m
ASR
site
−3.54
{10.20
}1.51
3{1.502
}0.51
5[0.45
0]30
0{>
1000
}0.33
0.57
{0.54}
[0.48]
0.60
{0.55}
[0.46]
0.62
{0.72}
[0.45]
0.43
{0.43}
[0.85]
64.5
{43.2}
[9.9]
88.5
{0.0}[55.4]
y=
2000
mdo
wn-
drift
ofASR
−1.60
{−6.04
}1.45
9{1.347
}0.52
5{0.459
}20
0{>
1000
}0.08
0.32
{0.44}
[0.24]
0.40
{0.42}
[0.32]
0.58
{0.66}
[0.56]
0.45
{0.43}
[0.53]
50.8
{29.5}
[8.5]
122.8{−
1.6}
[12.4]
Narrabe
eny=
2200
m0.37
{4.89}
4.51
2{3.854
}0.48
5{0.499
}30
{35}
0.00
0.17
{0.47}
[0.08]
0.57
{0.67}
[0.52]
0.78
{0.85}
[0.74]
0.73
{0.73}
[0.84]
117.4{74.5}
[42.2]
670.4{35.0}
[86.3]
y=
2600
m2.00
{6.77}
4.64
3{5.802
}0.49
3{0.539
}25
{15}
0.14
0.38
{0.48}
[0.30]
0.60
{0.66}
[0.45]
0.75
{0.82}
[0.39]
0.65
{0.67}
[0.52]
105.2{70.88
}[27.49
]43
1.4{29.2}
[49.2]
y=
3200
m2.26
{6.41}
5.36
9{5.111
}0.51
9{0.53
0}15
{15}
0.22
0.32
{0.49}
[0.16]
0.66
{0.68}
[0.52]
0.78
{0.81}
[0.81]
0.75
{0.71}
[0.88]
114.2{63.2}
[43.7]
689.8{33.7}
[106
.4]
195M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
constant equilibrium condition suffices for Gold Coast the application of aconstant value may not be generally appropriate. This assertion is furthersupported by the work of Wright et al. (1985), Harley (2009) and Harleyet al. (2011b) who noted significant hysteresis in the Narrabeen system,whereby present morphodynamic state and shoreline position werefunctionally dependent on past forcing. Thus, following Wright et al.(1985) a new approach to shorelinemodelling is investigated here, incor-porating a dynamic equilibrium condition that is nowpermitted to evolvetemporally with the weighted antecedent dimensionless fall velocity.
Additionally, theDLT10model (Eq. (3))was shown to be relatively in-sensitive to the value of the exponent k (Davidson et al., 2010). This maybe partly because in the optimisation process k is inseparably linked to c,whereby a decrease in one variable is compensated for by an increase inthe other, and vice versa. The previous Gold Coast optimisations of kand complimentary work reported by Yates et al. (2009) indicate thatan appropriate exponent of wave height is approximately one. Yates etal. (2009) select E1/2 (∝H1) for example for the Pacific US sites. Thus,from this point forward the exponent on H is set to a value of 1.
Within themodelling framework outlined in DLT10 the magnitude ofthe shoreline response is regulated by cΩk, which is proportional to (H/T)k. This formulation predicts a faster response with increased wavesteepness. However, it is more intuitive and better aligns withobservations that the magnitude of the shoreline response will bedominantly proportional to the available incident deepwater wavepower (∝Ho
2T), where Ho is the deepwater wave height. This is becausewave power, being a product of wave energy and celerity, representsthe flux of wave energy to the nearshore zone. This approach is nowadopted.
The previous discussion identifies several areas where improve-ment to generic shoreline model performance can be achieved. Theresulting shoreline forecast model, hereafter referred to as ShoreFortakes the following form:
dxdt
¼ bþ c�P0:5 Ωeq tð Þ−Ω tð Þ� �
ð4Þ
Where the forcing of cross-shore movement of the shoreline is nowthe product of the incident wave power P (computed using linear wavetheory), and the free parameter c that is indicative of the response ratewith units of velocity per measure incident wave power (m/s [W/m]−0.5).Following Yates et al. (2009), c is separated into erosion (c−, whenΩ>Ωeq) and accretion (c+, when ΩbΩeq) components, in recognitionthat erosion–accretion are controlled by different processes in natureand are observed to proceed at differing rates. As inDLT10, b is a shorelinetrend component, which is independent of wave forcing and crudelyaccounts for trends in the shoreline resulting from processes that havenot yet been considered.
As a fundamental new development the disequilibrium term(Ωeq(t)−Ω(t)) now incorporates a time-varying equilibrium condition,Ωeq(t). Based on the premise that both shoreline and morphologicalchange are inter-related, we follow Wright et al. (1985) and defineΩeq(t) as theweighted average of the antecedent dimensionless fall veloc-ity:
Ωeq tð Þ ¼
XDΔt
j¼0
Ωj10−jΔt=ϕ
XDΔt
j¼0
10−jΔt=ϕ
ð5Þ
The key parameter in Eq. (5) is the ‘memory decay’ of the system ϕ,where the weighting reaches 10%, 1% and 0.1% of the instantaneousvalue at ϕ, 2ϕ and 3ϕ days prior to the current calculation point. Theindex j represents the number of data points in the wave forcing timeseries prior to the calculation point at time t (j=0) and Δt representsthe sampling interval (here in units of days) of this same series.
196 M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
D represents the total windowwidth of the weighted average also mea-sured in days. Here D is fixed at 2ϕ in order to reduce the number of freeparameters in themodel and simplify the optimisation of themodel. Thesensitivity of the model to this assumption is assessed later. The optimalϕ-value is a model free parameter, determined by a simple iterativeleast-squares optimisation.
Mathematically, the optimisation of free parameters in Eq. (4) canbe achieved by a simple one-step least-squaresmethodology. The prob-lemwith this simplistic approach is that if there is a trend present in theshoreline data — perhaps due to gradients in longshore transport —
then the model can reproduce this trend either in the b-term, or equiv-alently, by adjusting the relative magnitudes of c+/c−. Either solution(adjusting b or c+/c−) produces a model with equal skill, leading to in-consistent estimates for the key c+/c− coefficients, when applied to an-other location with different shoreline trends. To avoid this unwantedand spurious effect, optimisation of free parameters in Eq. (4) is insteadachieved in a two-stage least-squares scheme. Stage 1 effectively iso-lates shoreline trends associatedwith long-term gradients inwave forc-ing, (the second term on the R.H.S. of Eq. (4)), from shoreline trendsthat result from other processes, such as gradients in longshore trans-port, that are subsequently accounted for (albeit crudely for the presenttime) by the b-term in Eq. (4) (stage 2).
1) Stage one assumes that, (neglecting longshore gradients in sedi-ment transport), if there is no long-term trend in the wave forc-ing then there must also be a zero-trend in the shoreline series.Therefore, stage one proceeds by using the linearly de-trendedshoreline ⟨x(t)⟩ and forcing terms ⟨P0.5(Ωeq−Ω)⟩, where the an-gular brackets indicate a de-trending operation which preservesthe time-series mean. The forcing term on the R.H.S. of Eq. (4) iscomputed and separated into accretion and erosion terms as perEq. (6a and b), then numerically integrated (Eq. (7)).
yþ ¼ p0:5 Ωeq−Ω� �D E
yþ ¼ 0for
Ω<ΩeqΩ > Ωeq
ð6aÞ
y− ¼ 0y− ¼ p0:5 Ωeq−Ω
� �D E forΩ<ΩeqΩ > Ωeq
ð6bÞ
yþ ¼ ∫N−1ð ÞΔt
t¼0
yþdt and y− ¼ ∫N−1ð ÞΔt
t¼0
y−dt ð7Þ
HereN is the total number of values in the forcing series. An initial so-lution (denoted by the subscript i) is obtained in the following leastsquares scheme, which accurately and consistently (for different loca-tions) computes the relativemagnitude of c+ and c−, uninfluenced byany shoreline trend thatmay have been present in the original record,which is uncorrelated with the trends in the wave forcing:
1 Yþi;1 Y−
i;1
1 Yþi;2 Y−
i;2… … …1 Yþ
i;n Y−i;n
2664
3775
aicþici−
264
375 ¼
x1x2…xn
* +2664
3775 or ΑiBi ¼ Ci ð8Þ
Here n is the total number of shoreline positions in the measuredtime-series (typically nbN). Initial model free parameters can beobtained from least squares as follows:
minBi∈R
‖ΑiΒi ‖ ð9Þ
Here thematrix elements inBi belong to a set of real numbers (ℜ) andthe double vertical lines represent the mathematical norm. Eqs. (6)
and (7) may then be re-evaluated, this time using the original waveforcing series with any trend preserved, yielding a new matrix A.The newmatrixA is then combinedwith the initialmodel free param-etersBi fromEq. (9), to yield an initial shoreline prediction (xi) that in-cludes any effects of trends in wave forcing present in the originalwave data:
xi tð Þ ¼ ABi ð10Þ
Any trend present in xi is correlated with trends in wave forcing andreferred to later as bwave. Here, bwave is assumed to be driven bycross-shore sediment transport processes.
2) Stage two of the optimisation yields any additional trend to bwave
that exists in the original shoreline series. Physically this secondtrend term (b) potentially (but at this stage rather crudely) encap-sulates a number of additional and otherwise unaccounted forprocesses including longshore gradients in sediment transport.Utilising the initial modelled shoreline from Eq. (10), thus fixingthe relative magnitudes of accretion/erosion coefficients, modifiedcoefficients (a, b and c) can be obtained form least squares optimi-sation of the following equation:
1 t1 xi;11 t2 xi;2… … …1 t3 xi;n
2664
3775
abc
24
35
x1x2…xn
2664
3775 ð11Þ
Here, t is the relative timing of the surveys in seconds, starting att1=0, which need not be regularly spaced in time and b is the shore-line trend component that is independent of trends in wave forcing(this being encapsulated already in xi). The final model predictions(xm) and free parameters are given by combining coefficients fromthe two stages:
xm tð Þ1 t1 Yþ
1 Y−1
1 t2 Yþ2 Y−
2… … … …1 tN Yþ
N Y−N
2664
3775
cai þ abccþicc−i
2664
3775 wheret ¼ Δt 0;1;2…N−1½ � ð12Þ
HereΔt is thewavemeasurement time-step and the finalmodel freeparameters to be used in Eq. (4) are b, c+=cci
+ and c−=cci−. The total
measured shoreline trend in the data will be equivalent to bt=b+bwave.
4. Model application
4.1. Model hindcasts
Figs. 3 and 4 show themodel hindcast results of applying the newlyderived ShoreFor model to the six Gold Coast and Narrabeen datasets,respectively. With reference to the model performance statistics inTable 2 it can be seen that ShoreFor performswell at all six locations. Re-membering that Harley et al. (2011b) showed that, averaged over thewhole embayment, approximately 60% of the shoreline variabilitycould be attributed to cross-shore sediment transport processes atNarrabeen (and less in the region of the present measurements), thepresent (cross-shore) model consistently explains around 57–66% ofthe shoreline variability. Brier skill scores measured using the linear fitbaselines are consistently rated as ‘good’ (BSS=0.62–0.78)with the ex-ception of the shoreline measured down-drift of the artificial surfingreef (y=2000 m) at the Gold Coast (BSS=0.4).
In all cases there is varying but consistent improvement in the per-formance of ShoreFor relative to the alternative baseline DLT10 model.The application of ShoreFor to both the Gold Coast and Narrabeen sitesreveals a modest improvement to the DLT10 Gold Coast shoreline pre-dictions (BSS=0.43–0.47), but major improvement compared to the
New ShoreFor Model − Gold Coast
2000 2001 2002 2003 2004 2005 2006 2007180
200
220
240
Shorefor ModelData: y=−1000m
Sho
relin
e P
ositi
on (
m)
2000 2001 2002 2003 2004 2005 2006 2007
60
80
100
120
Shorefor ModelData: y=900m
Time (years)2000 2001 2002 2003 2004 2005 2006 2007
−20
0
20
40
Shorefor ModelData: y=2000m
Fig. 3. Application of the new ShoreFor model to the Gold Coast for locations y=−1000 m (a), 900 m (b) and 2000 m (c). Notice the only modest improvement to the modelhindcast for profile y=−1000 m compared to the DLT10 model (Fig. 2). The thickness of the data curve (grey) indicates the potential measurement error.
197M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
previously poor hindcast for the Narrabeen site (BSS=0.65–0.75).These results are mirrored in the ΔAIC scores, which are all >1 usingboth the linear and DLT10 model baselines and take due considerationof the relative number of free parameters (refer Fig. 2).
4.2. Model sensitivity to memory decay (ϕ)
Fig. 5 shows two representative examples of the iterative opti-misation of the memory decay parameter ϕ for the Gold Coast(y=−1000 m) and Narrabeen (y=2600 m). The impact of settingthe window width (D) to varying multiples of ϕ is also examined
New ShoreFor Mo
2004 2005 2006 200720
40
60
80Shorefor ModelData: y=2200m
Sho
relin
e P
ositi
on (
m)
2004 2005 2006 200720
40
60
80Shorefor ModelData: y=2600m
Time (y2004 2005 2006 200720
30
40
50
60
70Shorefor ModelData: y=3200m
Fig. 4. Application of the new ShoreFor model to the Narrabeen embayment for locationshindcast for Narrabeen at profile y=2600 m compared to the DLT10 model (Fig. 2). The th
here. Whilst the Gold Coast example shows minimal sensitivity to thisvariability, the Narrabeen tests show a peak which is at a longer periodfor the D=ϕ case. There is a trend for the optimum ϕ-value to decreaseas D⇒4ϕ, the curves converge at 2ϕ and 3ϕ and correlation begins todrop off at 4ϕ. The convergence at 2–3ϕ suggests that this is an appro-priate generic solution, without compromising model skill. Fixing D=2ϕ usefully minimises the amount of antecedent wave data that isrequired.
Fig. 6 shows optimisation curves for ϕ for all six alongshore loca-tions at the Gold Coast and Narrabeen. Peaks in the optimisation curvesoccur at very different time-scales for the contrasting Gold Coast and
del − Narrabeen
2008 2009 2010 2011
2008 2009 2010 2011
ears)2008 2009 2010 2011
y=2200 m (a), 2600 m (b) and 3200 m (c). Notice the improvement to the modelickness of the data curve (grey) indicates the potential measurement error.
100
101
102
103
104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7r2
Gold Coast (y=−1000m)
D=1.PhiD=2.PhiD=3.PhiD=4.Phi
100
101
102
103
104
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Phi (days)
r2
Narrabeen (y=2600m)
D=1.PhiD=2.PhiD=3.PhiD=4.Phi
Fig. 5.Model skill as a function of the memory decay function ϕ. Model skill is measured in terms of the squared correlation coefficient between the model hindcast (ShoreFor) anddata. Optimisation curves are shown from the Gold Coast (top) and Narrabeen (bottom). The impact of varying the windowwidth D at fixedmultiples ϕ from 1 to 4ϕ is investigated.
198 M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
Narrabeen beaches. The Gold Coast optimisation curve asymptoticallyreaches a maxima at values of ϕ>200 days. At this level of averaging,Ωeq is suppressed to a weak seasonal signal value close to thetime-series mean, which in turn explains why the previous DLT10model worked well at this site. In contrast, at Narrabeen optimal ϕ ofvalues 15–30 days are an order of magnitude less, indicating a muchfaster shoreline response at this site corresponding to the individualstorm frequency. These contrasting time-scales in the ‘memory’ of thesystem at each site clearly match the dominant modes of observedshoreline variability at the two sites (see Figs. 3 and 4).
Gold
r2
100
101
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Narr
r2
Phi 10
010
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 6. Model skill (r2) as a function of the memory decay parameter (ϕ) for all profiles testedecay-value at Narrabeen is of the order 15–30 days whereas Gold Coast values are an ord
Fig. 7a and b show calculatedΩeq time-series corresponding to theoptimal value for ϕ at each of the six locations. It is notable that at theGold Coast (Fig. 7a) there is a distinct difference in Ωeq(t) up-drift(y=−1000 m, ϕ> 1000 days) of the artificial surfing reef, comparedto those at (y=900 m, ϕ=300 days) or down-drift (y=2000 m, ϕ=200 days) of this structure. This likely demonstrates that the reef sig-nificantly modifies the dynamics of the adjacent and down-driftbeaches. Contrastingly, although optimum ϕ-values vary between 15and 30 days for the Narrabeen site, this results in only a very subtle var-iability in Ωeq(t) between locations.
Coast
102
103
y=−1000my=900my=2000m
abeen
(days)10
210
3
y=2200my=2600my=3200m
d. The shaded grey area represents range in values. Notice that the optimised memoryer of magnitude higher.
2000 2001 2002 2003 2004 2005 2006 20071
2
3
4
5
6
Equ
ilibr
ium
om
ega
Gold Coast
phi= 200 days, y= 2000mphi =300 days, y= 900mphi>1000 days, y= −1000m
2004 2005 2006 2007 2008 2009 2010 20111
2
3
4
5
6
Date
Equ
ilibr
ium
om
ega
Narrabeen
phi= 15 days, y= 3600mphi =20 days, y= 2600mphi=30 days, y= 2200m
a)
b)
Fig. 7. Optimised Ωeq time-series for the Gold Coast (top) and Narrabeen (middle).
199M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
4.3. Model validation
To provide a more rigorous test of the ShoreFor model's predictivecapability, the available datasets were alternatively calibrated usingthe first half of the dataset, and then validated using the remainingunseen data. The results of this analysis are presented in Table 2and Fig. 8 (Gold Coast) and Fig. 9 (Narrabeen). Visually, the modelvalidations continue to predict the shoreline behaviour well on theunseen data. This observation is supported by the statistics whereBSS are most frequently rated as ‘good’ or ‘excellent’. Notable excep-tions (rated as ‘fair’) are profiles close to, or down-drift of the artificialsurfing reef at the Gold Coast and profile y=2600 m at Narrabeen
Gold
2000 2001 2002 2003180
200
220
240
Model calibrationModel validationData: y=−1000m
Sho
relin
e P
ositi
on (
m)
2000 2001 2002 200340
60
80
100
Model calibrationModel validationData: y=900m
Time (2000 2001 2002 2003
−20
0
20
40
Model calibrationModel validationData: y=2000m
Fig. 8. ShoreFor calibration–validation te
where the general behaviour is well predicted but the trend is notwell captured from the calibration series. Validation ΔAIC scores arepositive in all cases when compared to either a linear or DLT10model.
5. Discussion
The results shown in the previous section have demonstrated thatsignificant improvements to shoreline predictions over time-scales ofseveral years can be achieved through the inclusion of a time-varyingequilibrium condition. The equilibrium condition adopted here is aweighted function of the antecedent dimensionless fall velocity andis therefore also indicative of changing morphodynamic state. These
Coast
2004 2005 2006 2007
2004 2005 2006 2007
years)2004 2005 2006 2007
sts for the all Gold Coast locations.
200 M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
findings complement previous studies by Wright et al. (1985) andHarley (2009) who observed significant hysteresis at the Narrabeensite.
The inclusion of a temporally varying equilibrium condition im-plicitly (rather than explicitly) introduces negative feedback in themodel, constraining excessive shoreline displacement. That is, high/low-energy antecedent conditions (associated with prior shorelineerosion/accretion) will suppress further erosion/accretion. In thecase of erosion, negative feedback is thought to relate to increaseddissipation resulting from both the flatter beach face and prominentoffshore bar formed by prior erosion. Conversely, when antecedentcalm conditions persist, the offshore bar welds to the beach face, re-ducing the available offshore sand reservoir and flux of sand to theshoreface.
The previous section showed that the memory decay parameter(ϕ) at the Gold Coast and Narrabeen differed by at least an order ofmagnitude, the Gold Coast responding predominantly at seasonaltime-scales and Narrabeen at individual storm frequency. Why isthis? We hypothesise here that the observed difference in ϕ relatesto the differing modal beach states of the two contrasting beaches,which in turn dictate the efficiency of sediment exchange betweenthe shoreface and the offshore sandbar, nearshore circulation pat-terns and the offshore distance of sandbars.
Sandbar welding to the beach face and rapid shoreline accretion be-gins with the intermediate transverse bar and rip (TBR) beach state andis the dominant characteristic of lower wave energy low-tide terrace(LTT) and reflective (R) beach states (Wright and Short, 1984). Wrightet al. (1985) showed that the modal state of Narrabeen straddled the in-termediate rhythmic bar (RBB) and TBR beach states. These very dynam-ic beach states are at the tipping point for a bar welding event, such thatbar welding will occur rapidly under constructive wave conditions. Fol-lowing bar welding, shoreline recession and offshore bar formationwillquickly follow a storm. It is this rapid shoreline response to changingwave conditions thatwe hypothesise leads to storm-frequencymemorydecay values (ϕ=15–30 days) at Narrabeen. Also significant is the factthat the modal beach state at Narrabeen (RBB and TBR) is generallyassociated with strong 2D horizontal (cell) circulation, which in turnis linked with the more efficient cross-shore exchange of sediment(Splinter et al., 2011). Finally, the steeper beach gradient at Narrabeen
Narra
2004 2005 2006 200720
40
60
80 Model calibrationModel validationData: y=2200m
Sho
relin
e P
ositi
on (
m)
2004 2005 2006 200720
40
60
80 Model calibrationModel validationData: y=2600m
Time (2004 2005 2006 200720
40
60
80Model calibrationModel validationData: y=3200m
Fig. 9. ShoreFor calibration–validation t
results in breakpoint bars forming closer to the shoreline, facilitating ef-ficient and relatively rapid sediment exchange between offshore barsand the shoreface. Similar up and down-state morphodynamic changewere reported by Ranasinghe et al. (2004) at nearby Palm Beach. Theyused a process model to provide a more detailed assessment of themechanisms by which up and down-state transitions occurred.
At the Gold Coast the same three factors contribute to less fre-quent exchanges of sediment between the shoreface and offshorebars. Firstly, the higher mean dimensionless fall velocity of approxi-mately 4 indicates that the modal beach state will be more dissipativethan Narrabeen. The modal beach state at the Gold Coast is longshorebar and trough (LBT)/RBB. Thus, the Gold Coast has to traverse morebeach states before bar welding occurs than Narrabeen. Secondly, thelower average beach gradient (Table 1) at the Gold Coast means thatbreak point bars form further offshore (>2 times further) than atNarrabeen, again slowing sediment exchange between the offshorebar and the beach. Finally, the Gold Coast frequently exhibits stronglongshore currents and 1D horizontal circulation that has been dem-onstrated to inhibit cross-shore sediment exchange (Splinter et al.,2011).
The morphological differences between the two sites are conceptu-ally summarised in Fig. 10, which shows how beach state and (impor-tantly) the rate of change of beach state varies with dimensionless fallvelocity. Here the central figure is based on that presented by Wrightet al. (1985). Fig. 10 further highlights the difference in the modalbeach state between the two sites. The Gold Coast example imageshows the LBT/RBB morphology that is characteristically observed atthis site. This morphology is associated with the flatter, less dynamicsection of the beach state transition curve (and larger range in dimen-sionless fall velocity). Notice that Narrabeen is positioned on the moreintermediate and dynamically changing (i.e. steeper slope) portion ofthe transition curve, with rapid transition between TBR/LTT and rhyth-mic bar morphology (RBB). The representative Narrabeen images showan example of a rapid recovery-period transition between RBB (March31, 2005), through TBR/LTT (April 13, 2005) to reflective (April 27,2005), in just 27 days, illustrating how dynamic this site is.
The dynamic nature of Narrabeen compared to the more dissipa-tive Gold Cast site is also illustrated in the response time coefficients(c±, see Table 2), which are a factor of three times greater. In spite of
been
2008 2009 2010 2011
2008 2009 2010 2011
years)2008 2009 2010 2011
ests for the all Narrabeen locations.
201M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
the large variability of response rate parameter (c) between sites,inter-site variability is low (approximately ±6% for hindcast tests)and likely relate to small local variability in incident wave height.
The two stage optimisation process (Section 2.4) permits consis-tent estimation of the relative magnitude of the erosion and accretionresponse rate coefficients which are unbiased by shoreline trends.Inspection of Table 2 shows that for both sites the ratio of c−/c+ areapproximately 0.5±0.02 for model hindcasts. This promises a furtherreduction in model free parameters although application of the modelto more sites will be needed to confirm the constancy of this ratio.
Replacing PwithΩ (and adjusting the exponent k=1 to preserve anequivalent exponent for H) was also investigated. The result was aminimal change in model skill at the Gold Coast site but a significantdecrease in model skill at Narrabeen. Thus, the adoption of incidentwave power to force shoreline change appears the appropriate choice.
5.1. Model limitations
The most fundamental limitation of the current version ofShoreFor results primarily from the key processes that have thusfar been neglected. Omission of the effects of longshore sedimenttransport gradients, complex sources and sinks of sediments, surgeand sea level change likely limit the probable useful prediction hori-zon of the present model to approximately a decade, beyond whichthe impact of these processes are likely to significantly degradepredictions. The present model is expected to perform best at ex-posed coasts where cross-shore sediment transport accounts for
Fig. 10. Beach state equilibria and rates of change (Wright et al., 1985). Notice also the examthese sites. The modal beach state for Narrabeen (RBB/TBR) is situated on the steeper and ddissipative (top-left panel) and less rapidly changing with dimensionless fall velocity. Thepanel, March 31, 2005), through TBR/LTT (right-middle panel, April 13, 2005) to R (bottom
the majority of the shoreline variability. Additional future challengesinclude extending the model for suitable application to macrotidalcoasts and areas complicated by complex geology and man-madestructures. However, that said, by its definition the model will al-ways equal or outperform a simple linear extrapolation of the shore-line (included in Eq. (4)) based on past observations, which iscurrently the standard practice for many coastal management au-thorities. Very deliberately, the formulation of this model lends itselfto further extension to deal with the present limitations outlinedabove. Importantly, being fundamentally an empirical model, confi-dence in broader application of the ShoreFor modelling approachoutlined here is currently limited to the specific parameter spacefor which it has been calibrated.Wider application of this model to nu-merous sites of varied environmental conditions (sediment density,grain size and wave climate) is now encouraged so as to fully explorethe relationship between model free parameters and these readilymeasureable environmental variables.
6. Concluding remarks
This contribution details the development and assessment of anew shoreline prediction model — ShoreFor — which is suitable forapplication in micro-/meso-tidal, energetic, exposed sandy beacheswhere cross-shore transport processes dominate. The model wasapplied at two contrasting test sites, where six years of hourlywaves and weekly shoreline observations are available for modeltesting. Model hindcasts using ShoreFor were shown to explain
ple images from Narrabeen and the Gold Coast illustrating the typical beach states atynamic section of the transition curve. The modal Gold Coast state (LBT/RBB) is moreNarrabeen images show a typical recovery period transition between RBB (top-right-right panel, April 27, 2005) in just 27 days, illustrating how dynamic this site is.
202 M.A. Davidson et al. / Coastal Engineering 73 (2013) 191–202
approximately 60% of the observed weekly to multi-year shorelinevariability, which constitutes the majority of the shoreline change at-tributed to cross-shore processes at the Narrabeen site, (Harley et al.,2011b).
The development and formulation of this approach serves to high-light the importance of hysteresis in shorelinemodelling,whereby futureshoreline positions can be strongly dependent on past hydrodynamicconditions. Indeed the ShoreFor model was shown to significantlyoutperform a prior model that excluded this effect.
The model includes a temporally varying equilibrium conditionbased on a weighted average of the antecedent dimensionless fallvelocity, an equivalent formulation to that proposed by Wright etal. (1985) for predicting beach state. The success of this methodolo-gy at the morphologically contrasting test sites indicates a stronginterdependence between beach state and shoreline change. Theweighting of the antecedent condition is controlled by a memorydecay parameter (ϕ), which is related to the dominant time-scaleof shoreline variability. In ShoreFor ϕ is a free parameter that is ob-jectively optimised by comparison with field data. The resultspresented here show that the dominant time-scale of shoreline var-iability occurs at storm period (15–30 days) at Narrabeen, an inter-mediate embayed beach. At the Gold Coast however, ϕ is an order ofmagnitude higher at this dissipative site, responding more slowly atseasonal time-scales. It is hypothesised that the different memorydecay rates observed at the two sites relates to the differing modalbeach states and the corresponding efficiency of cross-shore sedi-ment exchange between the shoreface and offshore sandbars.
The general success of this new shoreline model for hindcasting theobserved shoreline behaviour at two distinctly different open-coast sitessuggests that this approach may be suitable for broader application.
Acknowledgements
The authors would like to acknowledge the very significant con-tribution of Melissa Mole to this work, including the preparation ofdata and proof reading of this manuscript.
The Gold Coast City Council and Warringah Council are acknowl-edged and thanked for financial support to UNSW that enabled the col-lection of multi-year video-derived shoreline time-series at the GoldCoast and Narrabeen study sites, respectively. The QLD Department ofEnvironment and Resource Management (formerly QLD EPA) andNSW Manly Hydraulics Laboratory are similarly acknowledged for thesupply of wave data.
This research was funded by the Australian Research Council(Linkage Grant LP100200448) and our Project Partners NSW Officeof Environment and Heritage,Warringah Council, Gosford City Coun-cil and CoastalCOMS. Dr Davidson would like to thank the PlymouthUniversities' Marine Institute financial support for his sabbatical thatmade this work possible, and a special thanks to UNSW School ofCivil and Environmental Engineering for additional funding andhosting his visits to work with the Australian team. Finally, wewould like to thank our reviewers of this manuscript for their helpfuland constructive input.
References
Akaike, H., 1974. A new look at the statistical model identification. IEEE Transactions on Au-tomatic Control AC 19 (6), 716–723.
Avdeev, A.V., Goriounov, E.V., Lavrentiev, M.M., Spigler, R., 2009. A behaviour-orientedmodel for long-term coastal profile evolution: validation, identification, and prediction.Applied Mathematical Modelling 33 (10), 3981–3996.
Cowell, P.J., Stive, M.J.F., Niedoroda, A.W., Swift, D.J.P., De Vriend, H.J., Buijsman, M.C.,Nicholls, R.J., Roy, P.S., Kaminsky, G.M., Clevering, A.J., Reed, C.W., De Boer, P.L., 2003.
The coastal-tract (part 2): applications of aggregated modeling to lower-order coastalchange. Journal of Coastal Research 19 (4), 828–848.
Davidson, M.A., Turner, I.L., 2009. A behavioral template beach profile model forpredicting seasonal to interannual shoreline evolution. Journal of Geophysical Re-search 114 (F01020). http://dx.doi.org/10.1029/2007JF000888 (21 pp.).
Davidson, M.A., Lewis, R.P., Turner, I.L., 2010. Forecasting seasonal to multi-year shore-line change. Coastal Engineering 57, 620–629.
Dean, R.G., 1973. Heuristic models of sand transport in the surfzone. Proc. of the 1stAustralian Conf. on Coastal Eng., Engineering Dynamics in The Surf Zone, Sydney,Australia, pp. 209–214.
Gourlay, M.R., 1968. Beach and dune erosion tests, Delft Hydraulics Laboratory Reportm935/m936. Delft, The Netherlands.
Hanson, H., 1989. GENESIS: a generalized shoreline change model. Journal of CoastalResearch 5 (1), 1–27.
Harley, M., 2009. Daily to decadal embayed beach response to wave climate forcing.Unpublished PhD thesis, University of New South Wales, Australia, 159pp.
Harley, M.D., Turner, I.L., Short, A.D., Ranasinghe, R., 2009. An Empirical model of beachresponse to storms — SE Australia. 19th Australasian Conference on Coastal andOcean Engineering, Engineers Australia, Wellington NZ 16–18 September.
Harley, M.D., Turner, I.L., Short, A.D., Ranasinghe, R., 2011a. Assessment and integrationof conventional, RTK-GPS ad image-derived beach survey methods for daily to de-cadal coastal monitoring. Coastal Engineering 58, 194–205.
Harley, M.D., Turner, I.L., Short, A.D., Ranasinghe, R., 2011b. A re-evaluation of coastal em-bayment rotation: the dominance of cross-shore versus alongshore sediment trans-port processes in SE Australia. Journal of Geophysical Research 116 (F04033), 16.http://dx.doi.org/10.1029/2011JF001989.
Holman, R.A., Stanley, J., 2007. The history and technical capabilities of Argus. CoastalEngineering 54 (6–7), 477–491.
Horrillo-Caraballo, J.M., Reeve, D.E., 2010. A comparison of the performance of CanonicalCorrelation Analysis on sand and shingle beaches. Marine Geology 274, 120–134.
Karunarathna, H., Reeve, D.E., Spivack, M., 2009. Beach profile evolution as an inverseproblem. Continental Shelf Research 29, 2234–2239.
Kuriyama, Y., 2012. Process-based one-dimensional model for cyclic longshore barevolution. Coastal Engineering 62, 48–61.
Larson, M., Kraus, N.C., 1995. Prediction of cross-shore sediment transport at differentspatial and temporal scales. Marine Geology 126 (1–4), 111–127.
Miller, J.K., Dean, R.G., 2004. A simple new shoreline model. Coastal Engineering 51,531–556.
Nam, P.T., Larson, P.M., Hanson, H., Xua Hoan, L., 2011. A numerical model of beachmorphological evolution due to waves and currents in the vicinity of coastal struc-tures. Coastal Engineering 58 (9), 863–876.
Plant, N.G., Holman, R.A., Freilich, M.H., Birkemeier, W.A., 1999. A simple model forinterannual sandbar behavior. Journal of Geophysical Research 104 (C7), 15,755–15,776.
Ranasinghe, R., Symonds, G., Black, K., Holman, R., 2004. Morphodynamics of interme-diate beaches: a video imaging and numerical modelling study. Coastal Engineer-ing 51 (7), 629–655.
Różyński, G., 2003. Data-driven modelling of multiple longshore bar evolution and in-teractions. Coastal Engineering 48 (3), 151–170.
Ruessink, B.G., Pape, L., Turner, I.L., 2009. Daily to interannual cross-shore sandbar mi-gration: observations from a multiple sandbar system. Continental Shelf Research29, 1663–1677.
Short, A.D., 1985. Rip-current type, spacing and persistence, Narrabeen Beach Australia.Marine Geology 65, 47–71.
Splinter, K.D., Holman, R., Plant, N.G., 2011. A behaviour-oriented dynamic model forsand bar migration and 2DH evolution. Journal of Geophysical Research 116,C01020. http://dx.doi.org/10.1029/2010JC006382.
Sutherland, J., Soulsby, R.L., 2003. Use of model performance statistics in modellingcoastal morphodynamics. Proceedings of the International Conference on CoastalSediments, pp. 1–14 (CD-ROM Published by World Scientific Publishing and EastMeets West Productions, Corpus Christi, TX).
Turner, I.L., Aarninkhof, S.J.D., T.D.T., Holman, R.A., 2006. Coastal imaging applicationsand research in Australia. Journal of Coastal Research 221, 37–48.
Van Rijn, L.C., Walstra, D.J.R., Grasmeijer, B., Sutherland, J., Pan, S., Sierra, J.P., 2003. Thepredictability of cross-shore bed evolution of sandy beaches at the time scale ofstorms and seasons using process-based profile models. Coastal Engineering 47,295–327.
Wright, L.D., Short, A.D., 1984. Morphodynamic variability of surf zones and beaches —a synthesis. Marine Geology 56 (1–4), 93–118.
Wright, L.D., Short, A.D., Green, M.O., 1985. Short-term changes in the morphodynamicstates of beaches and surf zones; an empirical predictive model. Marine Geology62, 339–364.
Yates, M.L., Guza, R.T., O'Reilly,W.C., 2009. Equilibrium shoreline response: observations andmodelling. Journal of Geophysical Research 114, C09014. http://dx.doi.org/10.1029/2009JC005359.
Yates, M.L., Guza, R.T., O'Reilly, W.C., Hanson, J.E., Barnard, P.L., 2011. Equilibriumshoreline response of a high wave energy beach. Journal of Geophysical Research116, C04014. http://dx.doi.org/10.1029/2010JC006681.