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<p>Data assimilation is the process of integrating observational data into a time series of state estimates. For linear-Gaussian systems, an optimal state estimate always exists. This optimal solution both minimizes mean-square error and maximizes the Bayesian posterior distribution at every time step. If the observation model is nonlinear, then the posterior distribution is no longer Gaussian; the minimum mean-square error state estimate and the maximum <i>a posteriori</i> (MAP) state estimate may not coincide. Still, the MAP estimate may be sought using optimization or other means. Multiple Data Assimilation (MDA) is a recursive approach that has been found empirically to produce similar state estimates to optimization-based approaches. However, convergence has been difficult to prove analytically. In this work, we shed new light on MDA by deriving it from a homotopy function. From this viewpoint, MDA is not an optimization but an approximate numerical solution to an ordinary differential equation (ODE). Given this new ODE formulation, we propose a new adaptive-step MDA algorithm inspired by the Runge Kutta integration method. The new algorithm outperforms standard approaches on a chaotic system with highly nonlinear measurements. We conclude that MDA is not an exact MAP estimator, but it can still provide a good approximation of the MAP given reasonable local linearity of the observation model.</p>

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Multiple data assimilation as an approximate maximum a posteriori estimator

  • 期刊论文

  • 发表日期：

    2025年4月8日

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  • Kristen Michaelson,
  • Andrey A. Popov,
  • Renato Zanetti

摘要

Data assimilation is the process of integrating observational data into a time series of state estimates. For linear-Gaussian systems, an optimal state estimate always exists. This optimal solution both minimizes mean-square error and maximizes the Bayesian posterior distribution at every time step. If the observation model is nonlinear, then the posterior distribution is no longer Gaussian; the minimum mean-square error state estimate and the maximum a posteriori (MAP) state estimate may not coincide. Still, the MAP estimate may be sought using optimization or other means. Multiple Data Assimilation (MDA) is a recursive approach that has been found empirically to produce similar state estimates to optimization-based approaches. However, convergence has been difficult to prove analytically. In this work, we shed new light on MDA by deriving it from a homotopy function. From this viewpoint, MDA is not an optimization but an approximate numerical solution to an ordinary differential equation (ODE). Given this new ODE formulation, we propose a new adaptive-step MDA algorithm inspired by the Runge Kutta integration method. The new algorithm outperforms standard approaches on a chaotic system with highly nonlinear measurements. We conclude that MDA is not an exact MAP estimator, but it can still provide a good approximation of the MAP given reasonable local linearity of the observation model.