Stochastic Lagrangian trajectory model for drifting objects in the ocean

Abstract

The prediction of drifting object trajectories in the ocean is a complex problem plagued with uncertainties. This problem is usually solved simulating the possible trajectories based on wind and advective numerical and/or instrumental data in real time, which are incorporated into Lagrangian trajectory models. However, both data and Lagrangian models are approximations of reality and when comparing trajectory data collected from drifter exercises with respect to Lagrangian models results, they differ considerably. This paper introduces a stochastic Lagrangian trajectory model that allows quantifying the uncertainties related to: (i) the wind and currents numerical and/or instrumental data, and (ii) the Lagrangian trajectory model. These uncertainties are accounted for within the model through random model parameters. The quantification of these uncertainties consists in an estimation problem, where the parameters of the probability distribution functions of the random variables are estimated based on drifter exercise data. Particularly, it is assumed that estimated parameters maximize the likelihood of our model to reproduce the trajectories from the exercise. Once the probability distribution parameters are estimated, they can be used to simulate different trajectories, obtaining location probability density functions at different times. The advantage of this method is that it allows: (i) site specific calibration, and (ii) comparing uncertainties related to different wind and currents predictive tools. The proposed method is applied to data collected during the DRIFTER Project (eranet AMPERA, VI Programa Marco), showing very good predictive skills.

Appendices

Appendix 1

1.1 Parameter confidence bands

The advantage of using the maximum likelihood method is that all consistent solutions are asymptotically normally distributed, that is,

$$ \hat {\user2{\theta}}\rightarrow N_{k}\left(\hat{\user2{\theta}},\,\Upsigma_{\hat{\user2{\theta}}}\right), $$

(22)

where $N_{k}\left(\hat{\user2{\theta}},\,\Upsigma_{\hat{\user2{ \theta}}}\right)$ denotes the k-dimensional normal distribution with mean vector $\hat{\user2{\theta}}$ and covariance matrix $\Upsigma_{\hat{\user2{\theta}}}.$ The covariance matrix $\Upsigma_{\hat \theta}$ is the inverse of the Fisher information matrix, I_θ, whose (r, j)th element is given by

$$ i_{rj}=-\left. \frac{\partial^2\ell(\theta|x)} {\partial \theta _{r}\partial \theta_{j}}\right|_{\theta=\hat \theta}. $$

(23)

This result allows calculating parameter standard deviations and hence confidence bands straightforwardly.

Appendix 2

2.1 Autoregressive moving average

ARMA(r, q) processes are time dependent models specially suitable to explore temporal dependencies. These models can be mathematically expressed as

$$ y_{t}=\sum_{j=1}^{r}{\phi_{j} y_{t-j}}+\varepsilon_{t}-\sum_{j=1}^{q}{\theta_{j}\varepsilon_{t-j}} $$

(24)

with r autoregressive parameters $\phi_{1},\,\phi_{2},\,\ldots,\,\phi_{r},$ and q moving average parameters $\theta_1,\,\theta_2,\,\ldots,\,\theta_q.$ The term $\varepsilon_{t}$ in Eq. 24 stands for an uncorrelated normal stochastic process with mean zero and variance $\sigma_{\varepsilon}^{2},$ and is also uncorrelated with $y_{t-1},\,y_{t-2},\,\ldots,\,y_{t-p}.$ Stochastic process $\varepsilon_{t}$ is also referred to as white noise, innovation term, or error term.