New links between invariant dynamical structures and uncertainty quantification

Abstract

This paper proposes a new uncertainty measure, appropriate for quantifying the performance of transport models in assessing the origin or source of a given observation. It is found that in a neighbourhood of the observation the proposed uncertainty measure is related to the invariant dynamical structures of the model. The paper illustrates the implementation of the proposed definition to quantify the performance of ocean data sets in the context of a real oil spill event in the Eastern Mediterranean in 2021.

Figures (7)

• Figure 1: Panels a), b) and c) display a graphical representation of two sequential observations and their forward time evolution. a) The initial observation at time $t_0$ is expressed by the red circle with initial condition ${\bf x_0}$. The final observed state ${\bf x^*}$ at time $t^*$ is referred to as the "target". In general, the evolution law linking these observations is unknown, but in our setting, it is approximated by Eq. \ref{['eq:saddle']}, a system with a hyperbolic fixed point at the origin with stable and unstable manifolds in blue and red, respectively. The forward evolution of ${\bf X_0}$ (the pink blob) to ${\bf X^*}$ (the light blue blob) according to this model is illustrated; b) a representation of $L_{UQ}$, in a domain beyond the neighbourhood $X_0$ for $\tau=3$, where singular features are identified aligned with the stable manifold (in blue); c) a representation of the stable manifold. Black arrows point out the forward time evolution for initial condition on the stable manifold towards the fixed point at the origin; Panels d), e) and f) display a graphical representation of two sequential observations and their backward time evolution, d) the final observation at time $t_1$ is expressed with the red circle at the position ${\bf x_1}$. Although both the origin and the model of this observation are unknown, its evolution is approached by Eq. \ref{['eq:saddle']}, a system with a hyperbolic fixed point at the origin with stable and unstable manifolds in blue and red respectively. The backward evolution of ${\bf X_1}$ (the pink blob) to ${\bf X^*}$ (the light blue blob) according to this model is illustrated; e) a representation of $L_{BUQ}$, in a domain beyond the neighborhood $X_1$ for $\tau=3$, where singular features are identified aligned with the unstable manifold (in red); f) a representation of the unstable manifold. Red arrows point out the evolution in backward time towards the fixed point for initial conditions on the unstable manifold. Black arrows point out the evolution forward time moving away from the fixed point. The color bar at the top indicates a color scale for $L_{UQ}$ and $L_{BUQ}$ in panels b) and e).
• Figure 2: A representation in the plane of $L_{BUQ}$ for the dynamical system \ref{['eq:saddle']} with target $x^*=(0.5,1 )$. a) $\tau=1$; b) $\tau=2$; c) $\tau=15$. The red line represents the position of the unstable manifold.
• Figure 3: For the system given by Eq. \ref{['eq:saddle']}, a) evolution of $L_{UQ}$ versus $\tau$ on the fixed point (black line) and on a point on the stable manifold (red line) with target ${\bf x}^*=(1.5,0.2)$; b) evolution of $L_{BUQ}$ versus $\tau$ on the fixed point (black line) and on a point on the unstable manifold (red line) with target ${\bf x}^*=(0.5,1)$.
• Figure 4: A graphical representation of the spills observed along the coastline of the Eastern Mediterranean and satellite observations matching the sources.
• Figure 5: Evaluation on the 16th of February 2021 of $L_{BUQ}$ using the target, ${\bf x^*}=(34.36^\circ \textrm{E}, 31.78^\circ \textrm{N})$ and of $M^{(b)}$ using $\tau=16$ days; a) $L_{BUQ}$ with the CMEMS global product; b) $M^{(b)}$ with the CMEMS global product; c) $L_{BUQ}$ with the CMEMS Mediterranean product; d) $M^{(b)}$ with the CMEMS Mediterranean product.