Forecasting seasonal to multi-year shoreline change

Coastal Engineering 57 (2010) 620–629

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Coastal Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o a s t a l e n g

Forecasting seasonal to multi-year shoreline change M.A. Davidson a,⁎, R.P. Lewis b, I.L. Turner c a b c

University of Plymouth, Drake Circus, Plymouth, Devon PL4 8AA, UK HR Wallingford Ltd, Howbery Park, Wallingford, Oxfordshire OX10 8BA, UK Water Research Laboratory, School of Civil and Environmental Engineering, University of New South Wales, Australia

a r t i c l e

i n f o

Article history: Received 29 May 2009 Received in revised form 13 January 2010 Accepted 4 February 2010 Available online 16 March 2010 Keywords: Forecasting Shoreline model Prediction Shoreline variability Monte Carlo simulation Australian Gold Coast

a b s t r a c t This contribution details a simple empirical model for forecasting shoreline positions at seasonal to interannual time-scales. The one-dimensional (1-D) model is a simplification of a 2-D behavioural-template model proposed by Davidson and Turner (2009). The new model is calibrated and tested using five-years of weekly video-derived shoreline data from the Gold Coast, Australia. The modelling approach first utilises a least-squares methodology to calibrate the empirical model coefficients using the first half of the dataset of observed shoreline movement in response to known forcing by waves. The model is then verified by comparison of hindcast shoreline positions to the second half of the observed shoreline dataset. One thousand synthetic time-series of wave height and period are generated that encapsulate the statistical characteristics of the modelled wave field, retaining the observed seasonal variability and sequencing characteristics. The calibrated model is used in conjunction with the simulated wave time-series to perform Monte Carlo forecasting of the resulting shoreline positions. The ensemble-mean of the 1000 individual fiveyear shoreline simulations is compared to the unseen shoreline time-series. A simple linear trend forecast of the shoreline position was used as a baseline for assessing the performance of the model. The model performance relative to this baseline prediction was quantified by several objective methods, including cross-correlation (r), root mean square (RMS) error analysis and Brier Skill tests. Importantly, these tests involved no prior knowledge of either the wave forcing or shoreline response. The new forecast model was found to significantly improve shoreline predictions relative to the simple linear trend model, capturing well both the trend and seasonal shoreline variabilities observed at this site. Brier Skill Scores (BSS) indicate that the model forecasts based on unseen data were rated as ‘excellent’ (BSS = 0.83), and root mean square errors were less than 7 m (≈ 14% of the observed variability). The standard deviations of the 1000 individual simulations from ensemble-averaged ‘mean’ forecast were found to provide a useful means of predicting the higher-frequency (individual storm) shoreline variability, with 98% of the observed shoreline data falling within two standard deviations of the forecast position. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In many routine long-term planning and policy decisions, coastal managers require estimates of future shoreline positions. Typically, this is achieved by fitting a linear trend to average annual erosion rates observed in the past, then forecasting N-years forward (with the size of N varying with geographical location), giving no consideration for the actual morphological processes forcing shoreline change. The intrinsic uncertainty of using a simple linear technique to model the stochastic nature of the climatic forcing is obvious (e.g. Douglas et al., 1998). At seasonal to interannual time-scales, complex process-based models have yet to yield reliable predictions (reviewed in De Vriend et al., 1993; Roelvink and Broker, 1993; Van Rijn et al., 2003). From the standpoint of their practical application to coastal management, such

⁎ Corresponding author. E-mail address: [email protected] (M.A. Davidson). 0378-3839/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2010.02.001

models may be considered to still be in a relatively early stage of development. Several simple empirical approaches have been more successful in modelling a variety of morphodynamic characteristics at the multi-year time-scale, including for example, sandbar position (Plant et al., 1999; Plant and Holman, 2001), shoreface gradient (Madsen and Plant, 2001) and shoreline position (Miller and Dean, 2004; Davidson and Turner, 2009). It is typical of such models that they are site specific and generally require an extensive data set for calibration purposes. An empirical model (similar to the latter genre) capable of forecasting future shoreline positions with a quantifiable degree of skill would provide a powerful tool for coastal management. Developing and verifying such a model are the focus of the present study. The starting point for the 1-D model presented here is the behavioural-template model previously proposed by Davidson and Turner (2009), hereafter referred to as DT09. The focus of the present work is to investigate whether the DT09 2-D profile model may be reduced to a more simplistic and efficient one-dimensional shoreline prediction model, whilst still retaining reliable shoreline predictions.

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The present model (like all profile and 1D shoreline models) assumes longshore uniformity and therefore negligible gradients in longshore sediment flux. Thus, this model is forced only with wave height and period and neglects wave angle effects which strongly affect longshore sediment fluxes. It is envisaged that later versions of the present model will include these longshore non-uniformities and result in a more versatile model. A Monte Carlo technique is used to run the new model for one thousand statistically similar realisations of the forcing wave climate, in order to test its forecasting ability. The procedure employs a piecewise, month-by-month, multivariate stationary simulation approach to generate realistic time-series of significant wave height (Hs) and wave period (Tp). Importantly, simulated data preserves the seasonality, distribution and wave sequencing found in the measured dataset. Uncertainties in model predictions are evaluated by a variety of techniques including cross-correlation of predictions and measurements (r), root mean square (RMS) error analysis and the Brier Skill Score (Sutherland et al., 2004), using a linear trend as the baseline for these tests. A description of the Gold Coast shoreline and wave time-series datasets used in this study is detailed in Section 2. Model description,

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assumptions and theory are given in Section 3. A description of the generation of synthetic time-series of wave data (Hs, Tp) and an explanation of the model skills tests are detailed in Section 4. Results of the model calibration, verification (i.e., hindcast) and forecast are presented in Section 5, and concluding remarks in Section 6. 2. Site and data description This work utilises five complete years of wave and shoreline data recorded at the Gold Coast, located on the south-east Pacific coast of Australia (Fig. 1). Beach sediments at this site have an average grain size (D50) and fall velocity (ws) of 0.25 mm and 0.03 m/s respectively. Energetic intermediate beaches whose bathymetry is defined by a persistent double-barred system typify the stretch of coastline under consideration here (e.g., Ruessink et al., 2007). The spring tidal range in this region is micro-tidal and of the order 1.8 m. The wave time-series consists of hourly measurements of wave parameters Hs and Tp from the years 2001 to 2006. These observations were obtained from a wave-rider buoy located approximately 2 km offshore of the study site, in 16 m water depth. Fig. 2 shows time-series of Hs, Tp and calculated dimensionless fall velocity (Ωm = Hs / wsTp).

Fig. 1. Location map of study site.

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Fig. 2. Time-series of Hs (top panel), Tp (middle panel) and Ωm (bottom panel) from 2001–2006.

Over the five-year duration of wave record the mean offshore significant wave height and spectral peak periods were 1.1 m and 9.4 s respectively. Maximum significant wave heights reached 5.9 m during a storm in early 2004.

Time-series of weekly measured shoreline positions (xm) were obtained from 10 min geo-rectified, time-averaged Argus images (Plant et al., 2007; Holman and Stanley, 2007; Holland et al., 1997). The shoreline position time-series (Fig. 3) were computed from a

Fig. 3. Shoreline decomposition. Top: measured shoreline position and trend. Bottom: seasonal/interannual variability (black), storm-frequency variability (grey).

M.A. Davidson et al. / Coastal Engineering 57 (2010) 620–629

longshore spatial-average of a 500 m stretch of coastline, thus serving to remove local and smaller-scale longshore variabilities (rips, cusps, etc.) in shoreline planform. The shoreline position was defined as the horizontal, cross-shore position at mean sea level. In order to reduce vertical errors in the video-derived shoreline position estimate due to wave set-up and swash oscillations (Plant et al., 2007), analysis of shoreline position was restricted to when Hs b 1 m. If the analysis coincided with the conditions where Hs N 1 m, the records were searched up to 3 days either side of this time until the condition Hs b 1 m was satisfied. A search window exceeding 3 days would mean that the shoreline estimate is too close to its neighbouring value to provide significant further information. Weekly shoreline estimates were found to be optimal as this sampling interval was rapid enough not to significantly alias the high-frequency variability in the actual shoreline at this site and long enough to work within the restraints of avoiding high energy conditions (Hs N 1 m). Given the restraints listed above it is estimated that the video survey of the horizontal shoreline time-series is accurate to within ±5 m of the estimate. The upper panel of Fig. 3 shows the measured shoreline timeseries together with the remaining shoreline trend of 1.3 m/a that is still present after the spatial averaging potentially indicating a residual longshore gradient in the sediment flux. A distinct seasonal succession is evident in the shoreline series, with erosion in the early to mid part of the year, followed by steady recovery periods in spring and early summer. In this regard, the period mid 2004 to early 2005 was uncharacteristic, with a relatively weak recovery period at the end of 2004 and low storm activity in the first three months of 2005. The lower panel in Fig. 3 shows the decomposition of the detrended shoreline time-series into low-frequency seasonal/interannual and higher-frequency storm components. Here a frequency domain Fourier filter has been used to low- (seasonal/interannual) and high- (storm components) pass the data with a cut-off frequency of 1/42 days−1. The division of variance in the shoreline time-series between the seasonal/interannual, trend (net accretion) and storm components is 55%, 35% and 10% respectively. Two conclusions may be drawn. First, seasonal/interannual variance contributes the highest variance to the shoreline time-series, and if the wave climate is correctly synthesised it is potentially possible to forecast this component. Second, the storm-frequency variability – of which the timing of individual storms is of course impossible to predict – comprises less than 10% of the variance at this site. For a more detailed summary of the wave climate and shoreline data sets including autoand cross-correlation analysis between the significant variables being modelled, the reader is referred to Davidson and Tuner (2009). 3. Model description As mentioned previously the starting point for this analysis is the 2-D behavioural-template DT09 profile model, which takes the following form: dzðx;t Þ n = RjΩo −Ωðt ÞjΩ ðt Þϕðx;t Þ dt

ð1Þ

In the above x is the cross-shore distance, z is the vertical displacement of the beach, t is time, Ω is the dimensionless fall velocity and ϕ is a shape function describing the form of the crossshore change including the formation of bars and steps in the profile. R, Ωo and n are model free parameters. R is a response rate parameter (m/s). Ωo represents the equilibrium dimensionless fall velocity. This formulation is attractive as the model forecasts more dramatic change for larger disequilibrium |Ωo − Ω(t)| (Wright et al., 1985) and larger/ steeper waves. Also intrinsic in the model is the effect of antecedent conditions, as Eq. (1) must integrate with respect to time to yield a solution for bed-level change.

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DT09 found that good model results were obtained using n = 2, although the complexity of the model made accurate optimisation of n difficult. Importantly for the present work, in the DT09 model the shape function ϕ was normalised such that the shoreline value (at x = 0) was unity. Thus, for x = 0 Eq. (1) can be simplified to: dzðt Þ n = RðΩo −Ωðt ÞÞΩ ðt Þ dt

ð2Þ

To obtain an estimate of the horizontal (cross-shore) movement of the shoreline, Eq. (2) is divided by the shoreface gradient: dxðt Þ R n ðΩ −Ωðt ÞÞΩ ðt Þ = dt tanβðt Þ o

ð3Þ

Eq. (3) can be further simplified by substituting Sunamura's (1984) parameterisation for tanβ as a function of Ω: −0:5

ð4Þ

tan βðt Þ = constΩðt Þ

Thus, Eq. (3) becomes: dxðt Þ k = cðΩo −Ωðt ÞÞΩ ðt Þ dt

ð5Þ

Where k = n + 0.5 and c = R/const. The DT09 model described by Eq. (1) was used by Davidson and Tuner (2009) to predict the de-trended variability in the shoreline measured at the Gold Coast. Here the model is extended to accommodate a linear trend in the data as follows: dxðt Þ k = b + cðΩo −Ωðt ÞÞΩ ðt Þ dt

ð6Þ

where b is the linear rate of net shoreline progradation or retreat. Since the shoreline trend is now included in the first term on the r.h.s. of Eq. (6), it is required that the second term which encapsulates the seasonal cross-shore redistribution of sediment produces zero trend. This fact allows the equilibrium value Ωo to be determined for any value of k such that: i = m dxðt Þ

∑

i=1

dt

 i=m
k = ∑ Ωðt Þ−Ωo;k Ωðt Þ = 0 i=1

ð7Þ

Here m is the total number of values in the calibration data set. For the purpose of practical calculation of the equilibrium value Ωo, simplification of Eq. (7) leads to: Ωo;k =

i = m k + 1 = 1 Ωi i=m ∑ Ωki i=1

∑i

ð8Þ

Efficient optimisation of the coefficients b and c (given k) in Eq. (6) may be achieved by integrating Eq. (6) and conducting a leastsquares analysis. Since this processes is highly computationally efficient it is easy to optimise for k by systematic trial-and-error. Integration of Eq.(6) yields: t

k

xðt Þ = a + bt + c∫0 ðΩo −Ωðt ÞÞΩðt Þ dt

ð9Þ

In the above, a is an offset (in meters), b is the linear trend coefficient (m/s) and c is the optimised response time coefficient (m/s) for the specified k. It should be noted that the trend term is likely to be dominated by longshore gradients in the sediment flux. No effort is made here to model the longshore gradient in sediment flux — this will form a logical extension in future work. The integral on the r.h.s.

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Fig. 4. Example time-series comparing Ωm (top) and a random selection of a Ωs time-series (bottom).

of Eq. (9) (termed X in the following) is consistent with the shoreline displacement due the seasonal cross-shore redistribution of sediment. X is computed numerically using the calibration wave data and then the unknown calibration coefficients (a, b and c) are obtained using a least-squares method to solve the following system of simultaneous equations: 2

1 61 6 4 ::: 1

t1 t2 ::: tm

3 X1 2 3 a X2 7 7⋅4 b 5 = 5 ::: c Xm

3 x1 6 x2 7 6 7 or A• B = C 4 ::: 5 xm 2

ð10Þ

Thus solving for the unknown coefficients B = A−1 • C. To quantify the skill of the above model, it is compared to a simple linear trend prediction of the following form: 2

1 61 4 ::: 1

3 t1   t2 7 a ⋅ = 5 ::: b tm

3 x1 6 x2 7 6 7 4 ::: 5 xm

the Ωm and Ωs time-series) cannot be predicted, the observed periods of higher storm activity in the first three months of each year (top panel), are also replicated in the simulated time-series (bottom panel). The precise timing of storm events cannot be forecast, but the statistical properties of each month can be replicated. Fig. 5 summarises the percentage occurrence of Ωm (measured) and all 1000 Ωs (synthetic) time-series. The distribution of simulated Ω-values is comparable to the measured Ω time-series. The Ωm data series has a mean of 4.18 and standard deviation of 1.87 compared to simulated values of 4.08 and 1.70 respectively. For this work, not only do the statistical properties of the measured wave field need to be encapsulated, but it is also essential to preserve the seasonal/interannual variability present in the measured series. Fig. 6 presents the results of autocorrelation analysis of Ωm, a single example synthetic series (Ωs), and the ensemble-average of all 1000

2

ð11Þ

4. Wave parameter simulation and model skills tests 4.1. Synthetic wave series The method of Borgman and Scheffner (1991) was used to generate one thousand, statistically similar 5-year Gold Coast wave records. Wave parameters Hs and Tp were simulated in order to generate corresponding synthetic dimensionless fall velocity timeseries Ωs. Fig. 4 shows a comparison between the measured (upper panel) and a typical synthetic (lower panel) dimensionless fall velocity time-series. Visual inspection of the two time-series shows that, though of course the timing of individual storm events (peaks in

Fig. 5. Histograms comparing Ωm (bars) and Ωs (line) time-series.

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Fig. 6. Auto-correlation analysis of measured (solid line) and example simulated Ω time-series (dotted line). Also shown is the autocorrelation of the ensemble-average of the 1000 simulated Ω time-series (grey line).

synthetic series (Ωs1000). This clearly demonstrates that seasonality is present in the measured series, and that this is replicated in the synthetic data with significant peaks at lags of 0, 52, 104, 156 and 208 weeks. The seasonality is most pronounced in the Ωs1000 autocorrelations, due to the fact that the shorter-term (stormfrequency) variability that is present in the individual simulations is effectively removed by the ensemble-averaging process. 4.2. Monte Carlo simulations The 1000 simulated time-series were used as boundary condition for the calibrated model described in Section 3 to generate an equivalent number of shoreline estimates. The resulting shoreline time-series were ensemble-averaged to produce a single shoreline prediction time-series. Additionally, a standard deviation time-series was generated reflecting the variability of the 1000 shoreline estimates around the ensemble-averaged shoreline position. Later (Section 5.3) the utility of this standard deviation time-series for providing error bounds for the shoreline estimate is examined.

The skill of the new 1-D shoreline forecasting model is assessed here using three statistical descriptors: 1. A simple linear regression between the measured shoreline series xm and the mean predicted series x, yielding a correlation coefficient (r). 2. The root mean square (RMS) error (in m), given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 〈ðx−xm Þ2 〉

" BSS = 1−

#

〈ð j x−xm j −ΔxÞ 〉 〈ðxb −xm Þ2 〉 2

ð13Þ

where Δx represents the measurement error and xb is the baseline measurement given by the linear trend. Here we explore the full range of possible measurement error from Δx = 0 to 5 m, with an intermediate value of 2.5 m used to indicate model performance. It should be noted that the definition of xb by a linear trend is a rather rigorous test; in morphodynamic analysis it is often the case that a single value equating to the initial value or record mean is alternatively used for this purpose. BSS have maximum values of 1.0 for a perfect prediction (allowing for the specified measurement error) and negative values may result if the predictions are worse than the benchmark value. Table 1 is included to aid interpretation of BSS (van Rijn et al., 2003). 5. Model calibration, verification and shoreline forecasts

4.3. Model skill scores

RMS =

Eq. (11), with consideration of additional measurement errors in xm time-series. The BSS is given as:

ð12Þ

where the angular brackets represent time-averaging over the assessment period. 3. A Brier Skill Scores (BSS — refer Table 1), which measures the ratio of improvement in accuracy of the forecast over a benchmark model, compared to the total possible improvement in accuracy (refer Sutherland et al., 2004). For the present work, the benchmark model is defined by the simple linear trend given in

In this section, half (0–2.5 years) of the measured shoreline timeseries is used to calibrate the model's free parameters, using the leastsquares methodology previously outlined in Section 3. These optimised parameters are then used together with the measured wave parameters to provide a hindcast of the remaining unseen (2.5– 5 years) of the measured shoreline series. The skill scores described in Section 4 are used to quantify the performance of the model hindcast against the unseen shoreline measurements. The optimised coefficients are then used to provide a true five-year forecast of shoreline movement, based upon the one thousand synthetic wave time-series. The skill of the forecast model is assessed by comparison of the unseen measured shoreline data. 5.1. Calibration The least-squares optimisation of the model's free parameters (a, b and c) in Eq. (9) was conducted for a range of values of the exponent k, utilising the first 2.5 years of the measured shoreline time-series. The

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Table 1 Definitions of values obtained from the Brier Skill Scores (van Rijn et al., 2003). Qualification

BSS

Excellent Good Reasonable/fair Poor Bad

1.0–0.8 0.8–0.6 0.3–0.6 0.3–0.0 b0.0

results of this analysis can be seen in Fig. 7 and Table 2. Interestingly, for this time period the maximum correlation (r = 0.76) and minimum RMS error (6.4 m) are coincident with k = 0.08 (i.e., k ≈ 0). The flatness of the optimisation curves combined with the fact that they are centred on k ≈ 0 implies that the model skill is relatively insensitive to the term Ωk and that it is possible to omit this term from the model, with little loss in skill, leading to a further simplification of Eq. (6):
 dxðt Þ — = b + c Ω−Ωðt Þ dt

ð18Þ

— Here Ω is the mean dimensionless fall velocity for the particular time-series being considered. However, it should be noted that the optimised k-value varies slightly (in the range 0–0.5) depending on the period of the data used in the optimisation process, so the Ωk term in Eq. (6) is retained. 5.2. Verification and hindcast The upper panel of Fig. 8 shows the measured dimensionless fall velocity that forces the model. In the lower panel, the corresponding measured (grey line), 0–2.5 year calibration (solid-black line) and 2.5–5 year hindcast (dotted–black line) shoreline positions are shown. For clarity, note that the 0–2.5 year calibration time-series is forced by the measured wave climate and is based upon the initial 2.5 years of measured and seen shoreline observations. In contrast, the 2–5–5 year verification hindcast is again forced by the measured wave climate, but for unseen shorelines during this second 2.5-year period. Inspection of Fig. 8 and Table 2 shows that the hindcast ability

of the model through the calibration period is classified as ‘excellent’ (BSS = 0.92, Δx = 2.5 m), as is also the case for the verification period (BSS = 0.83, Δx = 2.5 m). Notably, the model outperforms the linear trend on all relevant skills measures. Note that the methodology described above means that the linear trend component of the calibrated model (and the baseline linear extrapolation) is based only on the calibration period and not the entire record. 5.3. Forecast The optimised model was subsequently used to forecast a set of shoreline time-series using all 1000 synthetic Ωs time-series. For comparison to the observed shoreline data, ensemble-averaging all 1000 shoreline predictions produced a single forecast. Additionally, the standard deviations in the individual forecast time-series were used to derive confidence intervals for the forecast shoreline positions. A comparison between the measured shoreline and the ensemble-averaged forecast is shown in Fig. 9. The dotted lines represent ± one and two standard deviations from this forecast. As expected, the storm-frequency variability evident in the measured shoreline time-series is not retained by the forecast, as it is impossible to predict the specific timing of individual storm events. However, the seasonality in the measurements is faithfully reproduced. Skill scores based on only the last 2.5 years of the measured and forecast data (completely unseen) indicate that the correlation is significant at the 99% level (r = 0.56, p b 0.0001), RMS errors are b7 m and BSS are reasonable or better (N0.47), with average measurement error (Δx = 2.5 m) giving an ‘excellent’ value of BSS = 0.83. The above is a rather rigorous test of the forecast model for several reasons. Firstly, as was noted previously the second half of the measurement period (2004–2005) is uncharacteristic in terms of the seasonal succession at this site (Davidson and Turner, 2009). Indeed the correlation coefficient (r) increases to 0.73 if the full five-year record (including the initial 2.5 years used for calibration) is used to assess the model skill, rather than the final (unseen) 2.5 years only. Secondly, the prediction of shoreline movements was generated using the synthetic wave data only, with the measured data remaining completely unseen. Finally, the benchmark for the BSS test was the

Fig. 7. Optimisation of the exponent k in Eq. (9) using cross-correlation and root mean square error analysis in the comparing unseen measured and predicted shoreline series.

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Table 2 Summary of optimised model coefficients (a, b, c and k) and skills-score analyses. B = benchmark, C = calibration hindcast (seen waves and shoreline response), V = verification hindcast (seen waves, unseen shoreline response), F = forecast (unseen waves and shoreline response). r, R.M.S. error and BSS are the correlation coefficient, root mean square error and Brier Skill Scores respectively. Notice that the BSS has been quoted for measurement errors in the range of 0 to 5 m. Model

Test

a (m)

b × 10−8 (m/s)

c × 10−8 (m/s)

k

r

RMSE

BSS Δx = 5/2.5/0 m

Eq. Eq. Eq. Eq.

B C V F

207.61 214.68 214.68 214.68

19.50 9.28 9.28 9.28

N/A 222.93 222.93 222.93

N/A 0 0 0

0.45 0.76 0.50 0.56

8.74 6.34 7.62 6.96

N/A 0.99/0.92/0.67 0.97/0.83/0.54 0.99/0.83/0.47

(11) (9) (9) (9)

linear trend, and as was previously highlighted, this is a more stringent criterion than the alternative use of a single-point initial value or time-series mean. The confidence intervals of ± one and two standard deviations (1SD/2SD) from the ensemble-averaged forecast provide a useful means of predicting the storm-frequency variability around the seasonal forecast. 70% of the measurements fall within 1SD of the forecast, whilst 98% fall within 2SD of the forecast. In this example the standard deviation is ≈7 m, approximately 14% of the observed variability. Thus, although the model cannot predict the precise timing of the shoreline response to individual storms, the potential magnitude of storm response can be quantified. It should be noted that due to the constant fall velocity used in this formulation a model of equivalent skill could have been formulated around Hs / Tp in place of Ω. However, the original formulation with Ω remains attractive at the present time, as some consideration of the local sediment characteristics is likely to be an important inclusion when considering the model's application to a range of sites. It is also worthy of note that the optimisation for the Gold Coast is relatively insensitive to the precise value of the exponent k with optimal values very close to zero. This implies that the key driver for the model is the disequilibrium stress component (Ωo − Ω(t)) and not — the multiple Ωk. Furthermore, with k ≈ 0, Ω0⇒Ω , thus the threshold

value of dimensionless fall velocity separating shoreline recession (erosion) and progradation (accretion) is close to the time-series — mean Ω = 3.8 (or equivalently Hs / Tp = 0.125 m/s). Interestingly, similar formulations, replacing Ω with offshore wave steepness Hs / Lo (Lo is the deepwater wavelength) are significantly less skilful with calibration BSS values (Δx = 2.5) falling from 0.92 (using Ω or Hs / Tp) to 0.88 (using Hs / Lo). Nonetheless, threshold values of wave steepness (Hs / Lo) separating erosion and accretion for this site emerging from the optimisation process are approximately 0.006. As yet the present model has not been applied to sites other than the Gold Coast. It remains to be tested at sites of differing sediment and wave climate characteristics, but at the present time it is envisaged that the model will outperform the linear extrapolation methodology at locations where the seasonal shoreline signal is strong. It is noted that the current methodology summarised by Eq. (9) converges to a linear extrapolation as the seasonal signal diminishes, so it is likely to provide equivalent or superior predictions, relative to the linear extrapolation method. 5.4. Limitations A clear weakness of the present model is that the only inclusion of the potential effects of longshore gradients in sediment flux is via a

Fig. 8. Top: measured dimensionless fall velocity. Bottom: model calibration (black line) and verification (dotted line) hindcasts. The measured shoreline position is shown in grey.

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Fig. 9. Ensemble-average of 1000 model forecasts (black line) and measured shoreline position (grey line). Plus and minus one and two standard deviations of the model forecast is also shown (dotted lines). 98% of all the measured data fall within two standard deviation of the model forecast.

rather simplistic linear trend. The necessary assumption for the present work is that any longshore sediment transport gradients will remain constant over the duration of the forecast. This assumption will almost certainly constrain the prediction horizon of the present model formulation, although it is currently not possible to quantify this further. An extension of the current work in progress involves nesting the present model within a one-line shoreline model (e.g. Hanson, 1989), to better account for potential longshore gradients in sediment transport. A second limitation of the present model is that, by its empirical nature, a reasonably long time-series of wave and shoreline data is required to calibrate the models free parameters; here 2.5 years proves adequate. In order to appropriately characterise a site, it is anticipated that minimum sampling frequency for waves and shorelines will need to be of the order of daily and weekly, respectively. Weekly sampling of shorelines is labour intensive using conventional survey methods, with coastal video systems providing a more practical means of obtaining these data (Davidson et al., 2007; Holman and Stanley, 2007; Kroon et al., 2007; Smit et al., 2007; Turner and Anderson, 2007). The model performance is also highly dependent on the accurate statistical characterisation of the wave field. Here only a fairly short wave record (2.5–5 years) has been used to characterise the wave climate. It is recognised that such a short series would not adequately capture storms with return periods of greater than 1 per 5 years. This may limit the model's ability to predict the shoreline response to particularly extreme events. Thus, ideally much longer wave records should be used to characterise the wave climate. Additionally, longterm cycles (e.g. El Niño effects) and trends (N5 years) would demand more sophisticated wave climate modelling than those presented in the present contribution and this will be an important area of future research. Indeed the prediction horizon of the present model will probably be determined by the ability to predict future wave climate above all other factors.

As with any empirical model, application of the present model is only appropriate within the parameter space in which it is has been calibrated and tested. For example if wave heights exceed present day values at some time in the future then the model cannot be guaranteed to provide accurate estimates. However, increased storminess at a level which is similar to that used in the calibration and testing dataset is more likely to provide accurate predictions. The model (in its present simplistic form) will also be susceptible and unable to cope accurately with other physical changes in the boundary conditions in addition to the wave climate (for example beach replenishment or the construction of a breakwater) and would require recalibration after such an event. The 2-D profile model proposed by Davidson and Turner's (2009) to simulate the shoreline variability at the Gold Coast included both wave set-up and the effects of tidal displacement, with the results showing that inclusion of tides in particular can improve the prediction of shoreline change, even in a micro-tidal environment. Tides effectively distribute the impact of a storm over a broader intertidal region, with the effect that increased tidal range generally reduces the response of the shoreline contour. However, attempts to parameterise this effect within the current 1-D modelling framework did not lead to a significant improvement in model skill. 6. Concluding remarks A one-dimensional shoreline prediction model has been formulated, based on a simplification of the 2-D behavioural-template model proposed by Davidson and Turner (2009). The model effectively encapsulates the shoreline trend, seasonal and interannual shoreline displacement, and outperforms a more traditional forecast of the shoreline position based on linear extrapolation. One thousand five-year synthetic wave time-series were generated to encapsulate the statistical properties of the measured wave field, and shown to faithfully reproduce the seasonal and interannual

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variability observed in the measured data. These synthetic series were used in a Monte Carlo simulation to drive the calibrated model and generate one thousand shoreline prediction time-series. The ensemble-average of these shoreline forecasts using unseen wave and shoreline data showed excellent model skill (BSS = 0.83 for Δx = 2.5 m, r = 0.56) and again outperformed a simple linear extrapolation of the shoreline position. Although it is impossible to predict the timing of individual storm events in multi-year predications, the standard deviation of the modelled time-series can be used to predict the potential impact of these events on the shoreline position. For the five-year time-series of forecast shoreline positions at the Gold Coast, 98% of the measured shorelines were within 2 standard deviations of the ensembleaveraged model forecast. The 1-D forecast model is likely to perform most effectively where there are strong seasonal/interannual variabilities in the shoreline time-series, with the skill of the model converging to that of simple linear projection as seasonality in the forcing wave climate diminishes. As it is currently formulated, the present model assumes that gradients in longshore sediment transport are negligible or constant, and thus can be encapsulated by a linear trend. Current work is in progress to extend the model to better account for potential longshore gradients in sediment transport. Acknowledgements The authors would like to acknowledge the contribution of Rob Holman for pioneering and supporting coastal video research that make this sort of modelling approach possible. We would like to acknowledge Adrian Pedrozo-Acuña for his assistance in the wave climate simulation. Mark Davidson would like to acknowledge the support of the UK Engineering and Physical Sciences Research Councils funding in the RFPEBL research program. Gold Coast City Council funded UNSW to maintain and operate the northern Gold Coast Argus system. References Borgman, L.E., Scheffner, N.W., 1991. Simulation of Time Sequences of Wave Height, Period, and Direction. Technical Report DRP-91-2, 54. US Army Corps of Engineers, Washington. Davidson, M.A., Turner, I.L., 2009. A behavioural-template beach profile model for predicting shoreline evolution. Journal of Geophysical Research — Earth Surface 114, F1. doi:10.1029/2007JF000888.

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