Path planning in uncertain flow fields using ensemble method

Abstract

An ensemble-based approach is developed to conduct optimal path planning in unsteady ocean currents under uncertainty. We focus our attention on two-dimensional steady and unsteady uncertain flows, and adopt a sampling methodology that is well suited to operational forecasts, where an ensemble of deterministic predictions is used to model and quantify uncertainty. In an operational setting, much about dynamics, topography, and forcing of the ocean environment is uncertain. To address this uncertainty, the flow field is parametrized using a finite number of independent canonical random variables with known densities, and the ensemble is generated by sampling these variables. For each of the resulting realizations of the uncertain current field, we predict the path that minimizes the travel time by solving a boundary value problem (BVP), based on the Pontryagin maximum principle. A family of backward-in-time trajectories starting at the end position is used to generate suitable initial values for the BVP solver. This allows us to examine and analyze the performance of the sampling strategy and to develop insight into extensions dealing with general circulation ocean models. In particular, the ensemble method enables us to perform a statistical analysis of travel times and consequently develop a path planning approach that accounts for these statistics. The proposed methodology is tested for a number of scenarios. We first validate our algorithms by reproducing simple canonical solutions, and then demonstrate our approach in more complex flow fields, including idealized, steady and unsteady double-gyre flows.

Appendices

Appendix A: Hamiltonian and adjoint equations for the steady, stochastic double gyre flow

For the velocity field in Section 5.1, the Hamiltonian is expressed as:

$$\begin{array}{@{}rcl@{}} H &=&1+p_{1}(-\sin\left( \pi x\right)\cos(\pi y) + V\cos\theta \\ &&- \alpha\xi_{1} \sin(\pi (x+\delta\xi_{2}))\cos(\pi (y+\delta\xi_{3})) ) \\ &&+ p_{2}(\cos\left( \pi x\right)\sin(\pi y) + V\sin\theta\\ &&+\alpha\xi_{1}\cos(\pi (x+\delta\xi_{2}))\sin(\pi (y+\delta\xi_{3}))). \end{array} $$

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and the adjoint equations for the extended system are:

$$\begin{array}{@{}rcl@{}} \dot{p}_{1} = -\frac{\partial H}{\partial{x}} &=& p_{1}\pi\cos(\pi x)\cos(\pi y)\\ &&+ p_{1}\pi\alpha\xi_{1} \cos(\pi (x\,+\,\delta\xi_{2}))\cos(\pi (y+\delta\xi_{3})) \\ &&+ p_{2}\pi \alpha\xi_{1} \sin(\pi (x+\delta\xi_{2}))\sin(\pi (y+\delta\xi_{3}))\\ &&+ p_{2}\pi \sin(\pi x)\sin(\pi y), \\ \dot{p}_{2} = -\frac{\partial H}{\partial{y}} &=& -p_{1}\pi\sin(\pi x)\sin(\pi y)\\ &&- p_{1}\pi\alpha\xi_{1} \sin(\pi (x+\delta\xi_{2}))\sin(\pi (y+\delta\xi_{3})) \\ &&- p_{2}\pi \alpha\xi_{1} \cos(\pi (x\!+\delta\xi_{2}))\cos(\pi (y+\delta\xi_{3}))\\ &&- p_{2}\pi\cos(\pi x)\cos(\pi y). \end{array} $$

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Appendix B: Hamiltonian and adjoint equations for the unsteady, stochastic double gyre flow

For the velocity field in Section 5.2, the Hamiltonian is:

$$\begin{array}{@{}rcl@{}} H &=&1+p_{1}(-\sin\left( \pi f(x, t)\right)\cos(\pi y) +V\cos\theta\\ &&- \alpha\xi_{1} \sin\left( \pi f\left( x+\delta\xi_{2}, t\right)\right)\cos\left( \pi (y+\delta\xi_{3})\right)) \\ &&+ p_{2}\left( \alpha\xi_{1}\cos\left( \pi f\left( x+\delta\xi_{2}, t\right)\right)\sin\left( \pi (y+\delta\xi_{3})\right)\right.\\ &&\times\left.\left( 2a(t)\left( x+\delta\xi_{2}\right) + b(t)\right) + V\sin\theta \right)\\ &&+ p_{2}\cos\left( \pi f(x, t)\right)\sin(\pi y)\left( 2a(t)x + b(t)\right) \end{array} $$

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and the costate equations for the extended system are given by:

$$\begin{array}{@{}rcl@{}} \dot{p}_{1}= -\frac{\partial H}{\partial{x}}&=&p_{1}\pi\cos\left( \pi f(x, t) \right) \cos (\pi y)\left( 2a(t)x + b(t)\right)\\ &&+ p_{1}\pi\alpha\xi_{1} \cos\left( \pi f(x+\delta\xi_{2}, t)\right)\cos(\pi (y+\delta\xi_{3}))\\ &&\times \left( 2a(t)(x+\delta\xi_{2}) + b(t)\right)\\ &&+ p_{2}\pi \sin\left( \pi f(x, t)\right)\sin(\pi y)\left( 2a(t)x + b(t)\right)^{2}\\ &&+ 2p_{2} a(t)\cos \left( \pi f(x, t)\right)\sin (\pi y)\\ &&+ p_{2}\pi \alpha\xi_{1} \sin\left( \pi f(x+\delta\xi_{2}, t)\right)\sin(\pi (y+\delta\xi_{3}))\\ &&\times \left( 2a(t)(x+\delta\xi_{2}) + b(t)\right)^{2}\\ &&+ 2p_{2}\alpha\xi_{1} a(t) \cos\left( \pi f\left( x\,+\,\delta\xi_{2}, t\right)\right)\!\sin\!\left( \pi (y\,+\,\delta\xi_{3})\right),\\ \dot{p}_{2} = -\frac{\partial H}{\partial{y}} &=&-p_{1}\pi\sin\left( \pi f(x, t)\right)\sin(\pi y)\\ &&- p_{1}\pi\alpha\xi_{1} \sin\left( \pi f(x+\delta\xi_{2}, t)\right)\sin(\pi (y+\delta\xi_{3})) \\ &&- p_{2}\pi\cos\left( \pi f(x, t)\right)\cos(\pi y)\left( 2a(t)x + b(t)\right) \\ &&- p_{2}\pi \alpha\xi_{1} \cos\left( \pi f(x+\delta\xi_{2}, t)\right)\cos(\pi (y+\delta\xi_{3}))\\ &&\times \left( 2a(t)\left( x+\delta\xi_{2}\right) + b(t)\right) \end{array} $$

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At the final time, $t = \bar {t}_{f}$, the velocity components are:

$$\begin{array}{@{}rcl@{}} u_{f} &=& -\sin\left( \pi f(x_{f}, \bar{t}_{f})\right)\cos(\pi y_{f}) \\ &&-\alpha\xi_{1} \sin\left( \pi f\left( x_{f}+\delta\xi_{2}, \bar{t}_{f}\right)\right)\cos\left( \pi (y_{f} + \delta\xi_{3})\right) \end{array} $$

and

$$\begin{array}{@{}rcl@{}} v_{f} &=& \cos\left( \pi f(x_{f}, \bar{t}_{f})\right)\sin(\pi y_{f})\left( 2a(\bar{t}_{f})x_{f} + b(\bar{t}_{f})\right)\\ &&+ \alpha\xi_{1}\cos\left( \pi f\left( x_{f} + \delta\xi_{2}, \bar{t}_{f} \right) \right) \sin \left( \pi (y_{f} + \delta \xi_{3})\right)\\ &&\times\left( 2a(\bar{t}_{f}) \left( x_{f} + \delta\xi_{2}\right) + b(\bar{t}_{f}) \right) . \end{array} $$

Note that in the time-dependent case, the final time t _f is not known a priori. For the purpose of backward integration, the travel time at the end point is required to determine current. To overcome this difficulty, we introduce a guess $\bar {t}_{f} \in [t_{0}, t_{0} + T^{\max }]$ for the real travel time, where T ^max is the maximum time horizon. The guess for the final time is initialized using $\bar {t}_{f} = t_{0} + T^{\max }$, and then sequentially reduced, as needed, until a convergent solution is reached.