The problem of forecasting the behavior of a complex dynamical system through
analysis of observational time-series data becomes difficult when the system
expresses chaotic behavior and the measurements are sparse, in both space
and/or time. Despite the fact that this situation is quite typical across
many fields, including numerical weather prediction, the issue of whether the
available observations are “sufficient” for generating successful forecasts
is still not well understood. An analysis by

The ability to forecast the complex behavior of global
circulation in the coupled earth system
lies at the core of modern numerical weather prediction (NWP) efforts.
Successfully predicting such behavior requires both a good model of the
underlying physical processes as well as an accurate estimate of the state of
the model at the end of the analysis or observation window. When the model is
chaotic, even if it is known precisely, the accuracy of the prediction
depends on the accuracy of the initial state estimate. This is due to
sensitive dependence on the initial conditions, which was first identified by

Here we consider an idealized situation where a perfect dynamical model
describes the deterministic time evolution of a set of

This situation of high-dimensional dynamics and sparse measurements is
typical in the process of examining the consistency of observed data and
quantitative models of complex nonlinear systems: from functional nervous
systems to genetic transcription dynamics, among many other examples

To clarify these ideas we refer to the observability study given by

This lower bound was termed the critical number of measurements

Here we examine what can be done when

We now briefly discuss the concept of time delayed nudging; further details
can be found in

Measurements of the physical system are recorded during an observation window

The overall objective is to estimate the full model state

When the model is known precisely, a familiar strategy for transferring
information from the measurements to the model involves the addition of a
coupling or control or nudging term to Eq. (

This long-standing procedure, known as “nudging” in the geophysics and
meteorology literature, has been shown to fail when the number of
measurements at a given time falls below a critical value

It is therefore important to understand, for a given problem, whether

This idea provides the basis for the well-established technique in the
analysis of nonlinear dynamical systems, where this structure is employed as
a means of reconstructing unambiguous orbits of a partially observable system
(see, e.g.,

Note that the

In the estimation context, the time delays are used in a slightly different
way. Instead of reconstructing the topology of the attractor, they are used
to control local instabilities in the dynamics, which cause errors in the
analysis to grow. In other words,

Using this idea

The corresponding time delay model vectors

At each step of the integration of the controlled (nudged) dynamical
Eq. (

Furthermore, in the limit

Time delay nudging shares considerable overlap with incremental formulations
of strong constraint 4DVar

The main differences between the two methods are as follows.

Strong constraint 4DVar does not include the notion of a time delay or embedding dimension.

With the time delay method, the analysis is propagated in small increments

Time delay nudging uses truncated singular-value decomposition to regularize the solution, while strong constraint 4DVar uses a background term to perform Tikhonov regularization.

Furthermore, while we are currently working on unifying the motivating ideas behind time delay nudging with the variational action principle of weak constraint 4DVar, these and other related connections to 4DVar will be given in a subsequent paper. For the moment however, no additional theory will be introduced. Instead, we focus on its application to a core geophysical model: the shallow water equations.

We test our time delay nudging procedure through a series of

To simulate the conditions of a

It was previously shown by

We now describe the application of time delay nudging to a nonlinear model of
shallow water flow on a mid-latitude

As the depth of the coupled atmosphere ocean fluid layer (10–15

We have analyzed this flow using the enstrophy conserving discretization
scheme given by

Since these results were roughly consistent among the three resolutions
tested, we restrict our discussion here to the case where

Parameters used in the generation of the shallow water “data” for
the twin experiment. All fields as well as

Snapshots of shallow water flow on a

We now demonstrate that the time delay method is capable of reducing

The initial state

The coupling matrix

The time delay was selected to be

Synchronization error SE

Upper left panel: known (black), estimated (red) and predicted
(blue) values for the observed height values

Upper left panel: known (black), estimated (red) and predicted
(blue) for the observed

Upper left panel: known (black), estimated (red), and predicted
(blue) for the observed

Data assimilation results with

Synchronization error and known, estimated, and predicted height
values for

The effect of noise levels in the initial condition for the solution
of the model Eq. (

Initial positions for left panel

SE

Comparison of the estimated and predicted fields

The state was estimated by integrating the coupled differential
Eq. (

Short- and long-term synchronization error (Eq.

Since

The failure of predictions obtained with

Predictions were also calculated for

In the previous discussion it was suggested that reducing the coupling
strength would have a detrimental effect on the quality of the estimation
procedure and the resulting prediction. We investigate this now by performing
the same calculations as above with

This result demonstrates that proper choice of coupling is required. However, we have not developed a systematic way of choosing these values, and it is known from classical results on synchronization that the optimal choice depends on the number and distribution of observations. Furthermore, the fact that the height estimates appear to be rather accurate also emphasizes the point that, in a true experiment, the success of the assimilation procedure must be evaluated against the forecasts – not the analyses.

In addition, until now we have conveniently chosen to observe the height
field at all

Thus, even with time delays, it may not be possible to significantly reduce
the number of required height measurements. We remark however that the
overall space of parameters appearing in our study has not been thoroughly
explored. Additional refinement of the parameters

We now repeat the above calculations for

Another quite important source of observations about ocean flows is being
provided by position measurements

We monitor the positions of

We consider cases with

The consistency of the results improved by choosing the initial estimate to
only magnitude from the true solution, rather than in both
phase and frequency as was done above. Specifically, for the results reported
below the initial conditions of the data

In Fig.

Although we have not yet explored how to balance the number of drifters and
the number of height (or other) measurements, these preliminary results
suggest that positional data from drifters can be useful for improving the
observability of the system. In contrast to other approaches in which the
drifter data are used to directly interpolate the grid variables

The transfer of information from
measurements of a chaotic dynamical system to a quantitative model of the
system is impeded when the number of measurements at each measurement time is
below an approximate threshold

Here we have demonstrated how information in the time delays of the
observations may be used to reduce this requirement to about 30 %, in
which only the height fields need be observed. Moreover, it appears

Although all this has been done on a simplified model of shallow water flow,
implemented with only

Furthermore, we expect that this formalism will generalize to systems substantially larger than the one presented here, although we do not underestimate the numerical challenges involved in its extension to, say, the scale of operational NWP models. We also suspect this issue of insufficient measurements to be a critical limitation in our current ability to predict the behavior of complex, chaotic systems. Since such systems are quite typical in practice, these issues need to be examined with more realistic models.

Harking back to the introduction, we note that the report by

Data used in this article are
available at public data repository

This work was funded in part under a grant from the US National Science Foundation (PHY-0961153). Partial support from the Department of Energy CSGF program (DE-FG02-97ER25308) for Daniel Rey is appreciated. Partial support from the MURI program (N00014-13-1-0205) sponsored by the Office of Naval Research is also acknowledged. We would also like to thank the reviewers for their thorough reading and thoughtful suggestions that have helped significantly improve this manuscript. Edited by: Z. Toth Reviewed by: S. G. Penny and one anonymous referee