4.10 Sensitivity Analysis
4.10.1 Input Error Effects
A sensitivity analysis was carried out to assess model sensitivity to small changes in input wave data. Accurate measurement of wave characteristics is difficult therefore determining the effects of uncertainties on model predictions are important (Hanson, 1989). The model is centred around the calculation of longshore sediment transport rates which is subsequently used to calculate coastline changes. The Kamphius equation (eqn. 4.15) is used in this model, which uses three wave characteristics together with physical coastline parameters to calculate a longshore sediment transport rate.
Throughout the simulation period, wave characteristics are varied within the equation while other quantities remain constant. The effect of small changes to significant wave height at breaking (Hs), peak wave period (Tp) and relative wave angle at breaking (θb) are used as criteria for this sensitivity analysis. Additionally, beach slope (m) has also been included due to its dependence on the breaking wave height. Ignoring all constant values in the beach slope equation, it may be represented as:
m=H−
3 8
b (4.26)
Substituting this into the Kamphius equation yields:
Q=Hb138 Tp1.5sin0.6(2θb) (4.27) A numerical analysis was carried out to quantify the effects of small errors in wave input values. Errors were computed using a first order Taylor series approximation for each input variable. Combined error effects were also estimated by arithmetically multiplying the relevant input errors. An error of 10% was assumed such that the results may be compared to a similar analysis conducted by Hanson (1989) on the CERC formula. Table 4.3 details the values used for the sensitivity analysis.
Table 4.3 Wave characteristic values for sensitivity analysis.
Parameter Unit Value Change
Hs m 2.0 ±10%
Tp s 13.0 ±10%
θb o 15 ±10%
4.10.2 Model Stability and Accuracy
The coastline model uses an explicit solution technique as the new shoreline position is entirely dependent on calculated values at the previous time step. Explicit solution schemes are advantageous in the sense that programming is made easier along with boundary conditions being more easily expressible (Hanson, 1989). A major flaw however is the numerical stability, represented by the Courant Number (Rs) which is defined as the ratio between the time step and finite grid length (see eqn. 4.28).
The explicit scheme used in this model is second order correct however unlike implicit schemes, is not unconditionally stable (Dutykh, 2016). Hanson (1989) infers a maximum Courant Number of 0.5 when using explicit solution schemes for diffusion models while adding that numerical model results should be grid and time step independent.
Equation 4.28 arithmetically represents the Courant number, which states:
Rs=κ ∆t
∆x2 (4.28)
where κ is a diffusivity constant, ∆t represents the model time step and ∆x repre- sents the spatial grid intervals used in the model. Equation 4.20 may be alternatively expressed as:
∂y
∂t = sx Dc+Db
∂2y
∂x2 (4.29)
where sx is the coastal constant, represented as follows:
sx = ∂Qx
∂ϕ (4.30)
whereQx refers to calculated longshore transport rates andϕ refers to relative wave angles. Hence, we may represent diffusivity as:
κ= sx
Dc+Db = ∂Qx
∂ϕ(Dc+Db) (4.31)
This analysis focused on the effect of varying time steps and grid intervals on model stability and result accuracy. A hypothetical beach was initialised with constant wave parameters. The time step was varied with a constant grid interval to estimate the effects of a varying time step. The opposite was carried out to test the effect of varying grid intervals.
Chapter 5
Results & Discussion
This chapter presents the findings of this study. Comparison of observed and simulated wave climates are presented within this chapter together with the model calibration.
Outcomes of the sensitivity analysis are also shown. Finally, results of long-term simulations using varying sediment input schemes are presented.
5.1 Wave Climate Comparison
A comparison of the observed and simulated wave climate was carried out to assess similarity between datasets. Table 5.1 summarises both wave climates with selected statistical properties. The simulated climate covers all 101 iterations of 101 year wave sequences while the observed data corresponds to the 18 years of recorded data.
Table 5.1 Statistical properties of observed and simulated wave climates.
Property Observed Climate Simulated Climate
Mean Min. Max. Mean Min. Max.
Hs (m) 1.56 ± 0.51 0.01 8.5 1.70 ±0.60 0.1 30.24 Tp (s) 10.84 ±2.72 2.5 40.0 10.38 ± 3.93 1.05 20.0
D (o) 137.71 ±27.52 32.0 233.0 133.89 ± 30.42 30.0 210.0
Mean and standard deviation values for both wave climates show close resemblance regarding all wave parameters. A notable difference between the two wave climates is the maximum significant wave height (Hs) that occurs within the simulated wave climate. This value (30.24 metres) is sizeably larger than that of the observed wave climate (8.5 metres). Both values correspond to storm occurrences where the difference
may be attributed to the relative return periods of the storm events. The simulated data consists of 101 iterations of 101 year sequences, hence the likelihood of a 1:100 or 1:200 storm event occurring within one of the sequences is relatively high. Pringle et al.
(2015) substantiates this by stating a key feature of simulated waves is the inclusion of the correct number of extreme events. Another significant difference between wave climates is the observed maximum wave period (40.0 seconds) which is double that of the simulated climate (20.0 seconds). This value may be considered an outlier as it only occurs once in the entire dataset. The next highest value from the observed dataset is 22.0 seconds which varies from the simulated maximum by only 2.0 seconds.
Additionally, the same was found for the minimum wave height (Hs) for the observed wave data. The value (0.01 metres) was considered an outlier given its single occurrence, with the next highest value of 0.41 metres being a more common reading. An analysis of the wave climate distributions was then carried out.
Fig. 5.1 Probability distribution of wave heights based on direction.
Figure 5.1 shows the probability distribution of wave occurrences based on direction.
The simulated curve is representative of all 101 wave sequences. The two datasets show good correlation between probabilities across the entire wave spectrum. The observed data curve shows two distinct peaks at around 110 (P = 0.096) and 160 (P = 0.208) degrees separated by a distinct low around 130 (P = 0.070) degrees. This is replicated by the simulated data however not to the same extents. In comparison,
5.1 Wave Climate Comparison the simulated data yields probabilities of 0.081, 0.082 and 0.155 at 110, 130 and 160 degrees respectively. This may be attributed to the relative dataset sizes. Taking this into consideration, it may be said that both wave climates bear close resemblance in terms of directional occurrences of waves.
Fig. 5.2 Cumulative frequency distribution for observed and simulated wave heights.
Figure 5.2 shows the cumulative frequency distributions for the observed and simulated wave heights. The simulated curve (black line) corresponds to wave heights across all 101 iterations of wave sequences. It is evident that both are very similar regarding distributions of wave heights. The intersection point between curves at approximately 1.3 metres may be used to show the slight variations between datasets. Visibly, the simulated data contains a greater number of wave occurrences below 1.3 metres compared to the observed wave data. In contrast, the observed data exceeds the simulated data for essentially all wave heights above the intersection point until both graphs effectively asymptote at their peak cumulative frequencies.
Taking into consideration the statistical properties of both datasets together with a comparison of wave parameter distributions, it may be said that the simulated wave climate accurately assimilates the observed wave data. Use of the simulated wave climates for long-term model simulations should therefore provide an accurate indication of coastline behaviour.