Data assimilation transfers information from an observed system to a physically based model system with state
variables

In a broad spectrum of scientific fields, transferring the information
contained in

The conditional probability distribution

The action

With these conditions, the action takes a familiar form:

The conditional expected value

One interesting function

To approximate the integral

We take the the usual data assimilation technique

showing how to find a consistent path

demonstrating the importance of the number of measurements

explaining how to make systematic perturbation corrections
to

For nonlinear problems of interest, there may be many paths

Changing variables to

If the lowest action level

We now turn to an annealing method to find the path

The term in the action expressing uncertainty in the initial model state

The annealing method starts with the observation

For any

We begin the annealing process by choosing the initial

Because the search algorithm is an iterative process with potentially many
basins of attraction, it is not evident which minimum or saddle point we will
hit in this initial optimization. Accordingly, we actually start with

In the next step of the annealing process, the value of

This process is repeated as many times as desired, increasing

All of these optimizations, for each

As

Using properties of the

At the beginning of the annealing procedure when

The annealing procedure discussed above is different from standard simulated
annealing

The calculations used in the annealing process may seem formidable. We have
chosen

Action levels as a function of

We now present the results of a set of calculations on the Lorenz96

We performed a twin experiment in which we solved these equations, with

In the action, we selected

We also investigated using the IPOPT public domain numerical optimization
package

Data, estimated, and predicted time series for the Lorenz96 model

In Fig.

We think it important to note that if we had begun our search for the saddle
point paths

Another sense of why beginning a search at large values of

The real test of an estimation procedure in data assimilation is not accuracy
in the estimation of measured states and fixed parameters, but accuracy in
prediction beyond the observation window. The predictions require accurate
estimation of the unobserved model state variables at the end of the
observation window. Indeed, one can achieve good “fits” of observed
variables that lead to inaccurate predictions for

As this is a twin experiment, we show in Fig.

In the Lorenz96 equations, one usually has a single forcing parameter

In the twin experiments we presented noisy data from the known model as

Known and estimated forcing parameters for the Lorenz96 model at

We then place these signals as “data” in the action with the model taken as
Lorenz96

We also investigated, but do not display here, the action levels when the
parameter

We now have seen that a consistent smallest action level can be identified
via an annealing process, and the dependence of the action levels on the
number of measurements,

Action levels as a function of

List of important mathematical notations (in alphabetical order).

We turn back to the evaluation of the path integral for

In the form of the integral Eq. (

We have examined the path integral formulation of data assimilation
Eq. (

When measurement errors and model errors are distributed as iid Gaussian
noise, we have described an annealing method in the strength

We also explored the dependence of the action levels revealed through the
annealing method on the number of measurements

In previous work with variational principles for data assimilation, we are
unaware of any procedure such as our annealing method using

The relation of the annealing method to familiar 4DVar calculations

A similar annealing procedure in the importance of the model errors as
represented by

We have worked within a framework where the measurement errors and the model
errors in the data assimilation are Gaussian, with the inverse covariance of
the model errors taken to be of order

If the noise terms represented by the model errors are not Gaussian, one can
still use the annealing method to identify a path with the lowest action
level, but showing that perturbation theory about the path

In this paper, we do not address the typical situation where the number of
measurements actually available is less than that needed to allow the ground
level of the action

The results here justify the use of the variational approximation in data
assimilation, focus on the role of the number of measurements one requires
for accuracy in that approximation, and permit the evaluation of systematic
corrections to the approximation when the form of the action is
Eq. (

Partial support has come from the ONR MURI program (N00014-13-1-0205). We are very appreciative of the considered comments of the anonymous referees during the discussion period of this paper. Indeed, one of the first referee's questions led to the idea that the annealing approach could be useful in Monte Carlo estimations of the expected value path integrals. Edited by: Z. Toth; Reviewed by: two anonymous referees