Application of a stochastic differential equation to the prediction of shoreline evolution

Abstract

Shoreline evolution due to longshore sediment transport is one of the most important problems in coastal engineering and management. This paper describes a method to predict the probability distributions of long-term shoreline positions in which the evolution process is based on the standard one-line model recast into a stochastic differential equation. The time-dependent and spatially varying probability density function of the shoreline position leads to a Fokker–Planck equation model. The behaviour of the model is evaluated by applying it to two simple shoreline configurations: a single long jetty perpendicular to a straight shoreline and a rectangular beach nourishment case. The sensitivity of the model predictions to variations in the wave climate parameters is shown. The results indicate that the proposed model is robust and computationally efficient compared with the conventional Monte Carlo simulations.

Appendix 1: The theoretical framework for a general stochastic system

For a real field site where sediment transport pathways are parallel or shore-normal (both have significant components in two or three dimensions), the simplification made by the one-line alongshore model is unlikely to produce an accurate prediction for the variability of the coastal spatial configurations. In this case, a multi-dimensional deterministic numerical model can be incorporated into this modelling framework, involving a stochastic process of an initial state and some model parameters along these lines. For such a random dynamic coastal system, a more specific model that characterises a Markovian process is often adopted, which is obtained by a decomposition as the similar formulation with Eq. (12):

$$ {\dot{\mathbf{Y}}}(x,t) = {\varvec{\kappa}}({\mathbf{Y}},t) + {\mathbf{G}}({\mathbf{Y}},t){\mathbf{W}}(t) $$

(20)

where ${\mathbf{Y}} = (Y_{1} ,Y_{2} , \ldots ,Y_{{n_{\text{d}} }} )^{\text{T}}$ is the state vector, n _d is the dimension of the state space, κ(Y,t) is a function vector representing the effects of some form of drift, G(Y,t) is a function vector representing the effects of random diffusion, and W(t) is a n _e-dimensional Gaussian (or white) noise vector. Assuming that the white noise processes are mutually independent, the co-variance parameter matrix of W(t) will be a diagonal matrix with its diagonal terms equal to the variance of the white noise processes $W_{1} (t),W_{2} (t), \ldots ,W_{{n_{\text{e}} }} (t)$ and ${\mathbf{D}} = (D_{11} ,D_{22} , \ldots ,D_{{n_{\text{e}} n_{\text{e}} }} )$.

Equation (20) can also be written in the vector form of the standard Itô equation (see Soong 1973):

$$ {\text{d}}{\mathbf{Y}}(x,t) = {\varvec{\kappa}}({\mathbf{Y}},t){\text{d}}t + {\mathbf{G}}({\mathbf{Y}},t){\text{d}}{\mathbf{W}}(t) $$

(21)

with E{dW(t)} = 0 and E{[dW(t)]²} = 2Ddt.

The dynamical system described by Eq. (21) with the deterministic operators κ and G driven by some expressions of sediment transport, the evolutionary PDFs of the shoreline position Y, p(y,t), will satisfy the FP equation (Soong 1973; Gardiner 2004):

$$ \frac{{\partial p({\mathbf{y}},t\left| {{\mathbf{y}}_{0} ,t} \right.)}}{\partial t} = - \sum\limits_{j = v}^{{n_{\text{d}} }} {\frac{\partial }{{\partial y_{v} }}\left[ {\eta (y,t)_{v} p} \right]} + \frac{1}{2}\sum\limits_{u,v = 1}^{{n_{\text{d}} }} {\frac{{\partial^{2} }}{{\partial y_{v} \partial y_{u} }}\left[ {(GDG^{T} )_{uv} p} \right]} $$

(22)

The term (GDG ^T) _uv is given by:

$$(GDG^{\text{T}})_{uv}=\sum\limits_{k,l=1}^{{n_{\text{e}}}}{D_{kl}G_{uk} (y,t)G_{vl} (y,t)}\quad (u,v=1,2,\ldots,n_{\text{d}}),\;\;(k,l = 1,2, \ldots ,n_{\text{e}})$$

(23)

where G _uk and G _vl are components of G(y,t).

The theoretical framework described above is formulated into a stochastic shoreline evolution model to use for a n _d-dimensional SDE with a n _e-dimensional Wiener process. In these working examples, the state function vector is derived using the one-line model, which means that only longshore transport is considered to contribute to the shoreline changes, implying n _d = 1. For simplicity’s sake, only the characteristic wave height parameter is taken as a random parameter, implying n _e = 1.