# Modelling Cross-Shore Shoreline Change on Multiple Timescales and Their Interactions

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## Abstract

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## 1. Introduction

## 2. Model Training Sites Covering Different Wave Environments

#### 2.1. Narrabeen-Collaroy Beach, Australia

_{s}= 1.6 m and mean T

_{p}= 10 s). Additionally, the wave climate consists of larger waves (H

_{s}= 3 m) that originate from storm events and can hit the coastline in any direction. For example, in June 2007 an energetic storm sequence hit the coastline and caused roughly 35 m of shoreline retreat [38]. Furthermore, a small seasonal cycle is present with on average higher waves in the Australian winter and milder waves in the Australian summer months [38]. On larger timescales, effects of El-Nino Southern Oscillation (ENSO) can play a role as well [37,39,40]. In this study, nearshore wave timeseries at the 10 m depth contour are used.

#### 2.2. Nha-Trang Beach, Vietnam

_{s}) is 0.95 m, with an associated averaged peak period (T

_{p}) of 6.2 s. Typhoons typically occur on average 4–6 times per year between August and November. Intriguingly, typhoons were abundant in 2013, where the most extreme one occurred in November of that year [3]. The wave climate is also characterized by summer and winter monsoons, where the summer monsoons mainly consist of wind waves and the winter monsoons of swell waves. The winter monsoons (October to April), which do not occur at the same time as the typhoons, can generate waves up to 4.0 m, which can heavily affect shoreline change. During fall and winter (October to April), the mean H

_{s}is 1.2 m and T

_{p}is 6.8 s, while during spring and summer (May to September), the mean H

_{s}is reduced to 0.6 m, with a shorter T

_{p}below 5 s [30].

#### 2.3. Tairua Beach, New-Zealand

_{s}of 1.4 m, and can reach values of up to 6 m during storm events [43]. In the winter (southern hemisphere) of 2000 a series of huge storms occurred, which caused erosion of the shoreline of roughly 20 m [44].

## 3. Implementing Multiple Timescales and Links between Timescales in the Cross-Shore Shoreline Model

#### 3.1. Distinguishing Multiple Timescales

_{s}, T

_{p}) data is filtered using a running average filter with varying (enlarging) window sizes. The running average filter is used to allow for data gaps in the datasets. Per window, and hence timescale (i.e., 1 until the length of the timeseries, with steps of 1 day), a residual variance of the shoreline position and wave forcing timeseries is calculated. With residual variance of the filtered signals a temporal spectrum is constructed. In other words, we can plot the temporal spectrum as the (remaining) variance of all filtered signals as a function of the timescale. Dominant timescales are defined by the largest variance. In contrast to, for example, Fourier spectra, the linear superposition of all filtered signals is not equal to the raw signal since the filter function is shape preserving but not energy conserving. By dividing the filtered signals by a weighting value proportional to the window size of the running average filter, energy is conserved and the raw signal can be reconstructed through linear superposition. As a consequence, the energy (i.e., variability) for each filtered signal reduces, where the difference is largest for the largest timescales, because the window width increases with increasing timescales. A second temporal spectrum can be constructed from the resulting signals, which is used to construct timescale clusters by dividing the spectrum into bins (i.e., bands). These steps to distinguish timescales are summarized in Figure 2A–C.

_{s}and T

_{p}), an identical bin distribution is used such that corresponding timescale clusters are formed. Thereafter, the different timescale clusters in the wave forcing data (H

_{s}

_{,i}and T

_{p}

_{,i}, where i indicates the timescale cluster) are related to ones in the shoreline data on multiple scales. The interactions between the wave forcing and shoreline position timescale clusters on multiple timescales are based on three approaches: the direct forcing, the upscaling and downscaling approach. The combined model is referred to as Shoreline Forecast Multiple Timescales (SF-MT).

#### 3.2. Implementing Multiple Timescales

#### 3.2.1. Direct Forcing

_{s}

_{,i}and T

_{p}

_{,i}, where i indicates the timescale cluster) are related to corresponding timescale clusters in the shoreline position data (x

_{i}), following:

#### 3.2.2. Upscaling: The Long-Term Persistence of Short Timescales

#### 3.2.3. Downscaling: The Changing Response Efficiency of Short Timescales Due to Long-Term Shoreline Variations

_{i}; Equation (1)) and the large period between the two short high-intensity forcing events, the direct forcing approach will model two beach responses with the same erosional amplitude. Hence, no connection is present between the larger timescale shoreline variation and the modelled small timescale beach response to high-intensity forcing conditions.

_{j}(t); Equation (3)). This dynamic response factor represents the changing efficiency over time with which waves induce cross-shore sediment transport and is a function of the spatial separation between the shoreline and the more offshore sediment source (e.g., sand bar(s)). This spatial separation normally scales with the surfzone width imposed by the antecedent waves, with lower response rates for dissipating profiles (wide surfzone) due to the inefficient transfer of sediment between the more offshore region and the beach face. Conversely, as the more offshore sand supply is migrated closer to the shoreline (narrow surfzone; accreted beach), the sediment transport efficiency increases, facilitating faster response. Therefore, the dynamic response factor can be seen as an adjustment time scale and is a function of, for example, the current shoreline position, antecedent forcing conditions and morphological components such as sandbars. Therefore, within SF-MT the dynamic response factor has the shape of a large timescale shoreline signal, such that the response to a short high-intensity forcing event is higher (large sediment transport efficiency) when the beach (on a larger time scale) is accreted and lower (small efficiency) when the beach is already eroded.

_{j}(t)) (black-dashed) has the shape of the considered large timescale shoreline variation to account for a variable sediment transport efficiency, because beach response to small timescale high-intensity wave forcing events (red-solid) can depend on this larger timescale shoreline variation (i.e., the initial state of the beach). The figure shows that if the shoreline on a larger timescale is accreted (e.g., October 2013, high dynamic response factor), the relative (compared to the wave forcing, black-solid) shoreline response on a smaller timescale (red-solid) is large (higher sediment transport efficiency). Conversely, if it is eroded (e.g., January 2014, low dynamic response factor), the relative shoreline response is low (limited sediment transport efficiency). The resulting modelled shoreline timeseries is indicated by the dashed red line in Figure 4B.

#### 3.3. Model Calibration and Validation

_{i}and φ

_{i}) are determined using the wave forcing (H

_{s}

_{,i}and T

_{p}

_{,i}) and shoreline data (x

_{i}), where the subscript i indicates the multiple predictions from the direct forcing, upscaling and downscaling approaches. At the time of model validation, wave forcing data is used as model input only and together with the calibrated parameters shoreline predictions are generated. Table 1 presents the calibration and validation timeframes for all three datasets. Furthermore, note the difference in the percentages of data gaps in the shoreline position data (the wave forcing data is continuous for all three study sites).

#### 3.3.1. Calibration of the Downscaling Approach

_{j}(t), Equation (3)) has the shape of a larger timescale shoreline signal. However, this dependency poses a problem as during validation the model must generate shoreline predictions which are solely based on wave forcing (shoreline data is not available as model input). Hence, a different approach is needed to generate the dynamic response factor (i.e., the larger timescale shoreline signal). Note that the modelled shoreline signals from the direct forcing and upscaling approach are available before applying the downscaling approach. Therefore, those modelled signals will be used to capture the total shoreline response to come up with a shoreline signal. Subsequently, that shoreline signal will be filtered to generate shoreline timescale clusters that can be used as dynamic response factors.

**C**contains all the individual modelled shoreline signals (timeseries) with a distinct timescale that are generated using the direct forcing and upscaling approach, d the vector containing the measured shoreline data, the double vertical lines represent the mathematical norm and k the calculated vector containing values between the lower (zero) and upper (one) bound. Individual modelled shoreline signals (resulting from Equations (1) and (2) which are not important for the total modelled shoreline signal attain a value of zero, whereas the most important signals attain a value of one (i.e., the bounds determine the variability of the signal). The upper bound needs to be one as the optimization of the variability of the modelled shoreline signal to the shoreline data per timescale cluster is already performed at the time of the determination of the model-free parameters. Note that Σ

**C**·k thus represents a linear combination of the individual shoreline change timescale clusters. Moreover, this optimization procedure only determines the variability of the individual modelled signals to the total modelled shoreline signal. Before this procedure a different optimization process determined the values of c and φ per timescale cluster. Thus, a two-fold optimization process is used to come up with the total shoreline signal.

**C**only contains modelled shoreline signals that have a relatively high correlation with the corresponding measured timeseries. Modelled shoreline signals with a relatively low correlation (with the corresponding measured timeseries) result from a poor relation between the wave forcing and shoreline data and are therefore not used. The thresholds indicating a low/high correlation are determined through the fitting of a normal distribution to all correlation values per model improvement step. The thresholds indicating a high correlation are set to an optimized probability of exceedance of 90%, for both the direct forcing and upscaling approach. This ensures that most signals will be used as input for the linear least-squares solver, while only the poorest modelled shoreline signals are omitted. Subsequently, the resulting total shoreline signal is filtered and timescale clusters are formed following the same bin distribution as was used to determine the timescale clusters for the direct forcing and upscaling approach. These modelled shoreline timescale clusters are used as the dynamic response factor, such that all shoreline predictions in the validation phase, using the three modelling approaches, can be generated by the wave forcing only. Note that this procedure implies that shoreline predictions generated with the downscaling approach are partly based on shoreline predictions generated with the direct forcing and upscaling approach.

#### 3.3.2. Predicting the Total Shoreline Change

^{2}− N)/2 signals are generated). Note that the combination of modelled shoreline signals that will model the total shoreline change during the calibration phase (∑

**C**·k), is also responsible for modelling shoreline change during model validation (∑

**D**·k, where

**D**is the matrix containing the individual predicted shoreline signals for the validation phase).

## 4. Results

#### 4.1. Calibration

^{−7}(m/s)/(W/m)

^{0.5}). The SF-MT model (red) also captures the storm timescale to a certain extent, but yields a considerable increase in skill (BSS of 0.61 or an ‘excellent’ rating) by better capturing shoreline change on larger timescales (larger than the storm timescale). This occurs, for example, in 2009, where first the large erosion period halfway through 2009 is captured well by the SF-MT model and poorly by SF-ST, while the same occurs for the accretive period during the second half of 2009. Regarding the energetic storm sequence in June 2007 (see Section 2), neither model is capable of capturing the rapid shoreline recession. Overall, the SF-MT model yields a better fit to the data, compared to SF-ST, which seems to underestimate the amplitude of shoreline accretion/erosion most of the time. This is emphasized by the NMS error between the data and model result, which is 0.71 for SF-ST and reduces to 0.29 for the SF-MT model. This corresponds to a ‘fair’ and ‘excellent’ rating for SF-ST and SF-MT, respectively. The correlation coefficient is substantially larger for SF-MT as well (63%). The standard deviation and correlation plot per timescale (Figure 7A) shows that for storm/swell timescales (1–12 days) SF-ST has a higher standard deviation and a similar correlation. Hence, the smaller storm timescales are better captured by the SF-ST model. However, for SF-MT the standard deviation multiplied with the correlation is higher for timescales larger than 33 days (up to 2285 days). Hence, the dominant seasonal timescale is better captured by the SF-MT model, which contributes most to the model improvement. Moreover, the upscaling approach contributes considerably (73%) to the total shoreline signal (orange line in Figure 6A). The larger timescales are captured by the upscaling and downscaling (purple, Figure 7A) approaches whereas the smaller (storm/swell) timescales are captured by the downscaling approach as well. The direct forcing approach (blue) has a limited contribution to the total model result (7%, Figure 6A), compared to the upscaling (73%) and downscaling (20%) approach. The ∆AIC score (difference in AIC score between SF-ST and SF-MT) is larger than 1, which indicates that a considerable model improvement is acquired during the calibration phase.

^{−8}(m/s)/(W/m)

^{0.5}) only partially captures the seasonal timescale (the most dominant timescale, see Table 2 and Figure A1), the SF-MT model captures both the response to monsoons and the seasonal variation in wave data (i.e., the two dominant timescales in the wave data). The response to the energetic typhoon in November 2013 is not captured by SF-ST nor SF-MT. The NMS errors for SF-ST and SF-MT with the data is 0.31 (‘good’) and 0.13 (‘excellent’), respectively. Figure 7B shows that the standard deviation and correlation per timescale for SF-MT (red) are high and relatively uniformly distributed across the different timescales. This states that all timescales are well captured (except for the typhoons with a daily timescale). For SF-ST (black) the correlation is lower for all timescales except for the seasonal variation, where the correlation is similar. The standard deviation is lower/higher for timescales smaller/larger than 78 days. If both indicators are combined, it becomes clear that the largest contributor to the model improvement of SF-MT are the smaller timescales (i.e., the monsoons). Furthermore, note that an improvement is already made by using the upscaling approach only (orange line Figure 6B). However, in that case, only the response to the seasonal variation is captured. The response to monsoons (timescale of ≈ 20 days) is captured by the downscaling approach (purple). The relative contribution of each model improvement step indicates as well that the seasonal timescale (upscaling) is the most dominant (70%) in determining coastline evolution, followed by the monsoon response (downscaling, 23%). Figure 7B implies as well that the downscaling approach captures the smaller timescales (monsoons), while the larger timescales (seasonal variation) are captured by the upscaling approach. The ∆AIC score is larger than 1, which implicates that the larger number of calibration parameters in the SF-MT model is justified by a considerably better model fit (relative to SF-ST).

^{−7}(m/s)/(W/m)

^{0.5}) and SF-MT (red) model both capture shoreline change well for the dataset at Tairua and at first sight model differences are less pronounced than for the dataset at Narrabeen and Nha Trang. The NMS error indicates that there are differences: 0.51 for SF-ST and 0.36 for SF-MT. However, they both correspond to a ‘good’ rating, although the error is considerably lower for SF-MT. This difference is also emphasized by the BSS of 0.3, indicating that SF-MT is a ‘good’ improvement compared to SF-ST. The better model capability of SF-MT to capture shoreline change is emphasized as well in Figure 7C: both the standard deviation and the correlation for storm to monthly dominant timescales are higher for SF-MT. Those figures show as well that for the dominant seasonal to inter-annual timescales shoreline change is captured well by both models. However, there are certain moments in time where the SF-MT model outperforms SF-ST. This occurs, for example, at the end of 2006/beginning of 2007, where the accretion period is not well captured by SF-ST (Figure 5C). The rapid response to the series of huge storms that occurred in the winter of 2000 is captured reasonably by both models. From Figure 6C and Figure 7C becomes clear that the larger dominant seasonal to inter-annual timescales are modelled using the upscaling approach (orange, 57%), the smaller timescales are modelled using the downscaling (monthly timescale, purple, 27%) and direct forcing approach (storm/swell timescale, blue, 16%). Moreover, the ∆AIC score is larger than 1, which indicates that a considerable model improvement is acquired. Note that the correlation coefficient is higher for SF-MT as well (0.85 compared to 0.70 for SF-ST).

#### 4.2. Validation

## 5. Discussion

- Although camera-derived shorelines give a shoreline proxy, instantaneous and/or time averaged imagery is going to be affected by the role of tidal fluctuations, sea level anomalies and wave run-up. Moreover, sediments can be built up as a berm above the mean high water line and may not show up at all when using a single contour for the shoreline location. All those mechanisms/effects can have an influence on the shoreline location and are not accounted for in the model.
- Due to the linear interpolation of the shoreline location (for Nha Trang) and wave forcing parameters (i.e., H
_{s}and T_{p}) for all study sites to a single value uniformly spaced every 24 h apart, information is lost, which could induce errors when predicting shoreline change (e.g., storm and tidal aliasing effects). This data handling has a larger effect on the storm timescales compared to seasonal and inter-annual timescales (which were generally captured well by the model). Figure 7 shows that sub-weekly timescales are captured to a lesser extent compared to larger timescales (small standard deviation and correlation). Moreover, at these smaller timescales detailed morphodynamics can become critical for beach response, but they are not incorporated in SF-MT, and besides that, the equilibrium concept is at its limit for those small timescales. All these effects can be the cause for the relatively bad prediction skill for sub-weekly timescales. This is emphasized by the fact that the rapid response to the energetic (series of) storms, highlighted in Section 2, was not captured well by both models at Narrabeen and Nha Trang (and fairly reasonable at Tairua). - A second-order polynomial was used to detrend the raw shoreline position timeseries to remove timescales which are larger than the length of the dataset. However, it is possible that there is a relation between this second-order polynomial trend and how the beach responds to incoming wave forcing. For example, a strong erosive trend throughout the timeframe of the dataset would mean that the impact of monsoons become less as time progresses. This phenomenon is not implemented in SF-MT.
- As all study sites comprise embayed beaches, it is possible that longshore effects play a role due to, for example, long-term rotation of the beach, whereas the current model only accounts for cross-shore processes. At Narrabeen and Nha Trang, this mechanism is minimized to a certain extent as the shoreline location was determined by longshore averaging a certain stretch of the beach which is close to the center of rotation. Moreover, there could be an effect on the shoreline location of the angle of wave incidence (or sea level pressure variations as was shown by [50]) as it may also vary on different timescales. Those phenomena are not implemented in the model.

#### Timescale Interactions

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. ShoreFor (SF-ST), the Original Single-Timescale Model

^{+}) and erosional component (F

^{−}), because the accretionary and erosion responses are governed by different processes.

_{s}and T

_{p}are, respectively, the deep water-significant wave height and peak period and w

_{s}the sediment fall velocity.

_{s}

^{2}T

_{p}in deep water), ∆Ω the disequilibrium of dimensionless fall velocity (Ω

_{eq}− Ω) with Ω

_{eq}being a dynamic equilibrium term, Ω the instantaneous dimensionless fall velocity and σ

_{∆Ω}the standard deviation of the disequilibrium. The disequilibrium term (∆Ω) determines whether the coastline is accreting or eroding (plus and minus, respectively [17]) and dividing by the standard deviation of the disequilibrium makes sure that only the wave energy flux (P) and response rate parameter (c) determine the rate of shoreline change, rather than the magnitude of disequilibrium. The dynamic equilibrium term accounts for the fact that future shoreline positions can be strongly dependent on past hydrodynamic conditions [17] and can be, following [52], defined as follows:

_{n}the dimensionless fall velocity and φ the memory decay factor. A large memory decay factor (>> 100 days) generates a large timescale shoreline response (e.g., a seasonal variation), while a small decay factor (<100 days) produces a shoreline prediction where smaller (storm) timescales are dominant [18].

## Appendix B. Model Skill Assessment

_{m}is the measured shoreline and x the modelled shoreline. A NMSE of 0–0.3, 0.3–0.6, 0.6–0.8, and 0.8–1.0 is labeled as ‘excellent’, ‘good’, ‘fair’ and ‘poor’, respectively [18].

_{m}is the measured shoreline, x the modelled shoreline, δ the measurement error and x

_{b}the baseline model. In this paper, SF-ST is used as a baseline model. The triangle brackets indicate the mean. Positive BSS indicate a significant model improvement relative to this baseline model where values of 0–0.1, 0.1–0.2, 0.2–0.5 and 0.5–1.0 are labeled as ‘poor’, ‘fair’, ‘good’ and ‘excellent’, respectively. Note that the formulation of the BSS has been adopted after [58]. They advise the use of the skill formulation according to [57] in combination with a classification that is not adjusted for measurement error as is used here. A constant measurement error of 0.5 m is used, as for all sites, the shoreline location timeseries are extracted from video images.

_{σ}

^{2}the variance of the model or baseline residuals. Hence, it deals with the trade-off between the goodness of fit and model simplicity. If the difference between the baseline and model AIC (∆AIC) exceeds 1.0, a considerable model improvement is acquired [17]. In this study, SF-ST is used as a baseline model.

## Appendix C. Temporal Spectra of the Wave and Shoreline Data

**Figure A1.**Temporal spectra of the shoreline position (dotted), wave height (crosses) and wave period (diamonds), for the dataset at Narrabeen (

**A**), Nha Trang (

**B**) and Tairua (

**C**). The red circle indicates the most dominant timescale.

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**Figure 1.**Overview of the model training sites and positions of the camera stations from which high-resolution shoreline data is extracted. Left: Narrabeen and Colleroy beach embayment. Middle: the Nha Trang beach in Vietnam. Right: Tairua beach in New Zealand.

**Figure 2.**Multi-timescale implementation within SF-ST. (

**A**) Schematic overview of the filtering and reconstruction method of the raw (either shoreline or wave) signal (blue timeseries). Due to filtering, multiple signals emerge with different timescales (T

_{1}, T

_{2}, T

_{n}; black timeseries). By dividing each filtered signal by a value which is proportional to the window width of the filter function, the linear superposition of the resulting timeseries (red) does yield the raw signal. Subsequently, two spectra are constructed: one is used to identify dominant timescales (black, (

**B**), the other is used to construct timescale clusters (red, (

**C**)). (

**D**) shows an example of how bins (i.e., bands) are distributed and how corresponding timescale clusters are formed. The linear superposition of all filtered signals in a bin (in between two green vertical lines), result in one timescale cluster. The color-bar shows the number of filtered signals within a cluster, which is related to the variability of the filtered signals: the higher the variability, the lower the number of filtered signals in a bin, such that the total variability within a bin is constant. (

**E**) shows a schematic overview of the multi-timescale implementation in SF-ST using synthetic spectra. The signals corresponding to a timescale cluster in the wave forcing (H

_{s}, T

_{p}, indicated with 1, 2 and 3) and shoreline spectrum (indicated with I, II and III) are related in three ways: through direct forcing, upscaling and downscaling.

**Figure 3.**The upscaling approach: modelling the persistent effect of extreme forcing events on the longer-term state of the beach. The effect of the small timescale wave height timeseries (black-solid) on a larger shoreline position timescale (red-solid) is modelled using the envelope of the wave height timeseries (black-dashed), which creates the timescale link. A similar approach is followed for the wave period. The resulting modelled shoreline is given by the dashed red line.

**Figure 4.**The downscaling approach: modelling the effect of long-term shoreline trends on extreme event impacts. The effect of the larger timescales in shoreline variation on the efficiency with which smaller timescale wave forcing events induce cross-shore sediment transport is modelled (and shown in (

**A**)) by using a dynamic response factor (black-dashed), which has the shape of the larger timescale shoreline variation signal. (

**B**) Shows that if the shoreline on a larger timescale is accreted (e.g., October 2013, high dynamic response factor), the relative (compared to the wave forcing) shoreline response on a smaller timescale is large (higher sediment transport efficiency). Conversely, if it is eroded (e.g., January 2014, low dynamic response factor), the relative shoreline response is low (limited sediment transport efficiency).

**Figure 5.**Model calibration and validation results using SF-ST (black) and SF-MT (red) for the dataset at Narrabeen (

**A**), Nha Trang (

**B**) and Tairua (

**C**). Note that the measured data is indicated in blue. The black dashed line indicates the discrimination between the calibration and validation period.

**Figure 6.**Contribution of each model improvement step over time (left panels) for the dataset at Narrabeen (

**A**), Nha Trang (

**B**) and Tairua (

**C**). Note that the measured data is indicated in black and the direct forcing, upscaling and downscaling approach are indicated in blue, orange and purple, respectively. Right: the corresponding relative contribution of the three model improvement steps.

**Figure 7.**Standard deviation of the shoreline signals per timescale cluster (left) for the data (green), SF-ST (black) and SF-MT (red), considering the dataset at Narrabeen (

**A**), Nha Trang (

**B**) and Tairua (

**C**). The second column indicates the correlation coefficient between the model result and data for SF-ST (black) and SF-MT (red) for every timescale cluster and dataset. The third column shows a modelling score per timescale cluster and model improvement step. The score consists of the correlation coefficient (data model result) multiplied by the standard deviation (of the model result) for every timescale cluster and for every model improvement step. The blue-, orange and purple lines represent the direct forcing upscaling and downscaling approach, respectively.

**Figure 8.**Timescale interactions at Narrabeen (

**A**), Nha Trang (

**B**) and Tairua (

**C**) and the corresponding legend (

**C**). The diagonal represents signals modelled with the direct forcing approach where the black axes (from x to y) can be used to check which timescales are involved. For the upscaling approach (all patches in the upper left corner) the black axes can be used as well. The lower right corner of the grid represents signals generated with the downscaling approach. For those patches the red axes need to be used. The color indicates the percentage of those signals to the total modelled shoreline. A smooth function is used to visually highlight the most dominant timescale interactions.

**Table 1.**Calibration and validation timeframes of the Narrabeen, Nha Trang and Tairua dataset, as well as the percentages of data gaps in the shoreline position data (based on daily data).

Dataset | Calibration Timeframe | Validation Timeframe | Data Gaps in Shoreline Position [%] |
---|---|---|---|

Narrabeen | 1 August 2004/10 July 2010 | 11 July 2010/19 April 2015 | 25 |

Nha Trang | 27 July 2013/31 December 2014 | 1 January 2015/1 November 2015 | 20 |

Tairua | 2 January 1999 /31 December 2008 | 1 January 2009/31 December 2013 | 0 |

**Table 2.**Information of the Narrabeen, Nha Trang and Tairua dataset. Dominant timescales are defined as local maximums in the temporal spectra (see Appendix C).

Dataset | Timescale Bins | Dominant Timescale Shoreline Position [Days] | Dominant Timescale Wave Forcing [Days] |
---|---|---|---|

Narrabeen | 45 | 322 | 2–14 and 350 |

Nha Trang | 35 | 302 | 10–26 and 314–342 |

Tairua | 23 | 34 and 434–806 | 2–10 |

**Table 3.**Model skill during the calibration and validation phase, for the SF-ST and SF-MT model, at all three sites, using four model skill indicators.

Site | Indicator | Calibration | Validation | ||||
---|---|---|---|---|---|---|---|

SF-ST | SF-MT | % | SF-ST | SF-MT | % | ||

Narrabeen | R | 0.52 | 0.85 | 63 | 0.45 | 0.62 | 38 |

NMSE | 0.71 | 0.29 | −59 | 0.83 | 0.61 | −27 | |

BSS | - | 0.61 | - | - | 0.26 | - | |

∆AIC | - | >1 | - | - | <1 | - | |

Nha Trang | R | 0.83 | 0.94 | 13 | 0.86 | 0.89 | 3 |

NMSE | 0.31 | 0.13 | −58 | 0.63 | 0.26 | −59 | |

BSS | - | 0.62 | - | - | 0.61 | - | |

∆AIC | - | >1 | - | - | <1 | - | |

Tairua | R | 0.70 | 0.85 | 21 | 0.60 | 0.65 | 8 |

NMSE | 0.51 | 0.36 | −29 | 0.71 | 0.58 | −18 | |

BSS | - | 0.3 | - | - | 0.18 | - | |

∆AIC | - | >1 | - | - | <1 | - |

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## Share and Cite

**MDPI and ACS Style**

Schepper, R.; Almar, R.; Bergsma, E.; de Vries, S.; Reniers, A.; Davidson, M.; Splinter, K. Modelling Cross-Shore Shoreline Change on Multiple Timescales and Their Interactions. *J. Mar. Sci. Eng.* **2021**, *9*, 582.
https://doi.org/10.3390/jmse9060582

**AMA Style**

Schepper R, Almar R, Bergsma E, de Vries S, Reniers A, Davidson M, Splinter K. Modelling Cross-Shore Shoreline Change on Multiple Timescales and Their Interactions. *Journal of Marine Science and Engineering*. 2021; 9(6):582.
https://doi.org/10.3390/jmse9060582

**Chicago/Turabian Style**

Schepper, Rob, Rafael Almar, Erwin Bergsma, Sierd de Vries, Ad Reniers, Mark Davidson, and Kristen Splinter. 2021. "Modelling Cross-Shore Shoreline Change on Multiple Timescales and Their Interactions" *Journal of Marine Science and Engineering* 9, no. 6: 582.
https://doi.org/10.3390/jmse9060582