A Method for Assimilating Lagrangian Data into a Shallow-Water-Equation Ocean Model

Abstract

Lagrangian measurements provide a significant portion of the data collected in the ocean. Difficulties arise in their assimilation, however, since Lagrangian data are described in a moving frame of reference that does not correspond to the fixed grid locations used to forecast the prognostic flow variables. A new method is presented for assimilating Lagrangian data into models of the ocean that removes the need for any commonly used approximations. This is accomplished by augmenting the state vector of the prognostic variables with the Lagrangian drifter coordinates at assimilation. It is shown that this method is best formulated using the ensemble Kalman filter, resulting in an algorithm that is essentially transparent for assimilating Lagrangian data. The method is tested using a set of twin experiments on the shallow-water system of equations for an unsteady double-gyre flow configuration. Numerical simulations show that this method is capable of correcting the flow even if the assimilation time interval is of the order of the Lagrangian autocorrelation time scale (TL) of the flow. These results clearly demonstrate the benefits of this method over other techniques that require assimilation times of 20%–50% of TL, a direct consequence of the approximations introduced in assimilating their Lagrangian data. Detailed parametric studies show that this method is particularly effective if the classical ideas of localization developed for the ensemble Kalman filter are extended to the Lagrangian formulation used here. The method that has been developed, therefore, provides an approach that allows one to fully realize the potential of Lagrangian data for assimilation in more realistic ocean models.

A hybrid grid/particle filter for Lagrangian data assimilation. II: Application to a model vortex flow

“…In each case, the dependence of the errors on the size of the ensemble is explored. In addition, two different sampling times are used; the first is given by t assim = 50 t and corresponds to values used for the results presented in Figure 3, whilst in the second case the assimiltion time is increased three-fold to t assim = 150 t. We recall that our main motivation for formulating the hybrid filter was to address the problem of filter divergence encountered in the work of Kuznetsov et al (2003), and Salman et al (2006). The tests with t assim = 150 t, therefore, provide a direct test of our method in this parameter range.…”

“…Note, however, that in contrast to the methods of Kuznetsov et al (2003), Ide et al (2002), and Salman et al (2006), there is no need to solve the stochastic ordinary differential equations for the observation state vector x D since these equations have been replaced by the set of partial differential equations given by Equation (2).…”

“…As a result, this has made Lagrangian data an attractive tool for use in ocean forecasting. On the other hand, this very mechanism is also responsible for the abrupt divergence of most filters that have recently been formulated for Lagrangian data assimilation (Ide et al (2002), Kuznetsov et al (2003), Salman et al (2006)). To resolve this problem, we have presented, in part I of this paper, a new nonlinear filter specifically formulated for Lagrangian data assimilation.…”

“…The first method we will compare against is the perturbed observation EnKF of Evensen (1994) and Burgers et al, (1998). This filter was recently used in the Lagrangian data assimilation studies of Salman et al (2006), (2008) and is known to show filter divergence if the assimilation time-interval is too large. This arises because, whilst the method retains the fully nonlinear evolution operator in computing the transitional PDF between filtering steps, at analysis, the EnKF invokes the assumption of a Gaussian PDF as in the Extended Kalman Filter (EKF).…”

“…The Ensemble Kalman filter Salman et al (2006Salman et al ( , 2008 employed the EnKF to assimilate Lagrangian data into a shallow-water equation ocean model. As with the hybrid method, Equation (1) is used to obtain the model forecasts of our flow state vector.…”