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A Causation-Based Computationally Efficient Strategy for Deploying Lagrangian Drifters to Improve Real-Time State Estimation

Erik Bollt, Nan Chen, Stephen Wiggins

Abstract

Deploying Lagrangian drifters that facilitate the state estimation of the underlying flow field within a future time interval is practically important. However, the uncertainty in estimating the flow field prevents using standard deterministic approaches for designing strategies and applying trajectory-wise skill scores to evaluate performance. In this paper an information measurement is developed to quantitatively assess the information gain in the estimated flow field by deploying an additional set of drifters. This information measurement is derived by exploiting causal inference. It is characterized by the inferred probability density function of the flow field, which naturally considers the uncertainty. Although the information measurement is an ideal theoretical metric, using it as the direct cost makes the optimization problem computationally expensive. To this end, an effective surrogate cost function is developed. It is highly efficient to compute while capturing the essential features of the information measurement when solving the optimization problem. Based upon these properties, a practical strategy for deploying drifter observations to improve future state estimation is designed. Due to the forecast uncertainty, the approach exploits the expected value of spatial maps of the surrogate cost associated with different forecast realizations to seek the optimal solution. Numerical experiments justify the effectiveness of the surrogate cost. The proposed strategy significantly outperforms the method by randomly deploying the drifters. It is also shown that, under certain conditions, the drifters determined by the expected surrogate cost remain skillful for the state estimation of a single forecast realization of the flow field as in reality.

A Causation-Based Computationally Efficient Strategy for Deploying Lagrangian Drifters to Improve Real-Time State Estimation

Abstract

Deploying Lagrangian drifters that facilitate the state estimation of the underlying flow field within a future time interval is practically important. However, the uncertainty in estimating the flow field prevents using standard deterministic approaches for designing strategies and applying trajectory-wise skill scores to evaluate performance. In this paper an information measurement is developed to quantitatively assess the information gain in the estimated flow field by deploying an additional set of drifters. This information measurement is derived by exploiting causal inference. It is characterized by the inferred probability density function of the flow field, which naturally considers the uncertainty. Although the information measurement is an ideal theoretical metric, using it as the direct cost makes the optimization problem computationally expensive. To this end, an effective surrogate cost function is developed. It is highly efficient to compute while capturing the essential features of the information measurement when solving the optimization problem. Based upon these properties, a practical strategy for deploying drifter observations to improve future state estimation is designed. Due to the forecast uncertainty, the approach exploits the expected value of spatial maps of the surrogate cost associated with different forecast realizations to seek the optimal solution. Numerical experiments justify the effectiveness of the surrogate cost. The proposed strategy significantly outperforms the method by randomly deploying the drifters. It is also shown that, under certain conditions, the drifters determined by the expected surrogate cost remain skillful for the state estimation of a single forecast realization of the flow field as in reality.
Paper Structure (31 sections, 3 theorems, 20 equations, 8 figures)

This paper contains 31 sections, 3 theorems, 20 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A Causation-Based Information Measurement to Quantify the Information Gain by Deploying Additional Drifters
  3. State estimation with uncertainty
  4. Causation entropy: measuring the uncertainty reduction using information metrics
  5. Quantifying the information gain using a modified version of the causation entropy
  6. The optimization problems using the information measurement as the cost function
  7. The reanalysis scenario
  8. The real-time state estimation scenario
  9. Computational challenges in the direct optimization
  10. A Computationally Efficient Surrogate Cost Function
  11. Lagrangian descriptor: a computationally efficient trajectory diagnostic approach
  12. Developing a surrogate cost function with the help of Lagrangian descriptor
  13. Sequential strategy
  14. The entire procedure of deploying drifter observations
  15. The reanalysis scenario
  16. ...and 16 more sections

Key Result

Proposition 4.1

Given one realization of the drifter trajectories $\mathbf{X}(s\leq t)$, the filtering posterior distribution $p(\mathbf{U}(t)|\mathbf{X}(s\leq t))$ of Lagrangian data assimilation eq:cgns is conditionally Gaussian, where the time evolutions of the conditional mean $\boldsymbol\mu$ and the condition with $\cdot^*$ being the complex conjugate transpose.

Figures (8)

  • Figure 3.1: Procedures of deploying drifters in both the reanalysis and real-time forecast scenarios.
  • Figure 5.1: Comparison the information gain of different methods. Panel (a): using brute-force search algorithm to sequentially deploy $L_2=4$ drifters, one at each time. The contour shows the exact information gain computed from the relative entropy \ref{['Signal_Dispersion']}. The red dot shows the location of the newly added drifter, which is at the location corresponding to the maximum information gain. Panel (b): the true flow field at $t^*=5$, where the color map shows the amplitude $\sqrt{u^2+v^2}$. Panel (c): the surrogate cost computed from the expected Lagrangian descriptor using the flow field recovered from $L_1$ drifters. The large shading circles indicate the use of distance criterion, where the minimum distance is $1.5$ units here. Panel (d): information gain using different methods. The three green lines represent the information gain using the brute-force search but with different numbers of mesh grids (namely different spatial resolutions). Note that the information gain for the red line and blue dots will have slight shift up- or down-wards if a slight different minimum distance is used as the criterion. But the red line (all-at-once strategy) always stays above most ($\sim90\%$) of the blue dots and the sequential strategy slightly outperforms the all-at-once strategy.
  • Figure 5.2: Surrogate cost maps by sequentially deploying the $L_2=4$ drifters. In each column, the top and bottom panels have the same contour plot, representing the surrogate cost. But the bottom panel also includes the locations of the drifters. The existing drifters are represented by black dots. Each time one new drifter is deployed and it is indicated by the red dot. The surrogate cost maps given by the expected Lagrangian descriptor are similar, but the uncertainty triggers some differences. Such differences will change the locations to deploy new drifters, which then helps improve the information gain.
  • Figure 5.3: Determining the locations of discharging drifters in the real-time state estimation scenario. The parameters are the same as those in \ref{['Parameters_eddy_model']} except $\sigma_\mathbf{k}=0.125$. There are, in total, $L_1=10$ drifters existing in the field. Their trajectories from $t=0$ to $t=T=2$ are available. The additional $L_2=4$ drifters are discharged at $t=2$, aiming to improve the state estimation for a future period $t\in[2,2.5]$. Panel (a): the truth and the estimated time series of mode $(-3,-3)$. The cyan curve from $t=0$ to $t=2$ is the true signal (only the real-part is shown). The blue curve within the same interval is the posterior mean from Lagrangian DA (again only the real-part is shown). The blue shading area shows the two standard deviations from the mean, representing the uncertainty in the Lagrangian DA, where the standard deviation is the square root of the posterior variance of this mode. The uncertainty of the model ensemble forecast from $t=2$ to $t=2.5$ is shown in the blue shading area, where the posterior distribution at $t=2$ serves as the initial condition of running the forecast model \ref{['OU_process']}. The twenty thin blue curves from $t=2$ to $t=2.5$ are the ensemble members of the forecast trajectories, and the red curves and the red shading areas are the posterior mean and two standard deviations when the $L_2$ additional drifters are placed at the locations determined by the surrogate cost function at $t=2$. Panels (d) is a zoom-in illustration showing one of the twenty realizations. The associated time evolution of the posterior variance is shown in Panel (e). The contour plot in Panel (b) shows the map of the surrogate cost function computed from the expected Lagrangian descriptor \ref{['LD_VelocityBased_Expectation']}. The blue dots indicate the existing $L_1$ drifters, while the red dots are the locations of the $L_2$ additional drifters at $t=2$. Panel (c) shows the flow field at $t=2$ and the predicted trajectories of the drifters from $t=2$ to $t=2.5$.
  • Figure 5.4: Surrogate cost maps associated with each of the twenty realizations of the forecast flow field in the case presented in Figure \ref{['LD_forecast']}.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Proposition 4.1: Posterior distribution of Lagrangian data assimilation: Filtering
  • proof
  • Proposition 6.1: Posterior distribution of Lagrangian data assimilation: Smoothing
  • proof
  • Proposition 6.2: Sampling trajectories from posterior distributions
  • proof