The Lagrange multiplier method for combining observations and models (i.e., the adjoint method or “4D-VAR”) has been avoided or approximated when the numerical model is highly nonlinear or chaotic. This approach has been adopted primarily due to difficulties in the initialization of low-dimensional chaotic models, where the search for optimal initial conditions by gradient-descent algorithms is hampered by multiple local minima. Although initialization is an important task for numerical weather prediction, ocean state estimation usually demands an additional task – a solution of the time-dependent surface boundary conditions that result from atmosphere–ocean interaction. Here, we apply the Lagrange multiplier method to an analogous boundary control problem, tracking the trajectory of the forced chaotic pendulum. Contrary to previous assertions, it is demonstrated that the Lagrange multiplier method can track multiple chaotic transitions through time, so long as the boundary conditions render the system controllable. Thus, the nonlinear timescale poses no limit to the time interval for successful Lagrange multiplier-based estimation. That the key criterion is controllability, not a pure measure of dynamical stability or chaos, illustrates the similarities between the Lagrange multiplier method and other state estimation methods. The results with the chaotic pendulum suggest that nonlinearity should not be a fundamental obstacle to ocean state estimation with eddy-resolving models, especially when using an improved first-guess trajectory.

The most complicated, and probably most realistic, numerical models of the
ocean circulation are eddy-resolving ocean general circulation models

For the Lagrange multiplier method to be successful in state-of-the-art ocean
models, two major issues need to be addressed: (1) the high dimensionality of
the forward model and estimation problem, and (2) the nonlinearity of ocean
models at increasingly fine resolution. Research conducted by the ECCO
(Estimating the Circulation and Climate of the Ocean) Consortium

Due in part to the concerns raised about nonlinearity in simple models, the
method of Lagrange multipliers has rarely been applied to realistic models
over time windows longer than the eddy scale. For example, some studies
restricted the time windows to be short enough that unstable modes would not
grow too large

In this research, we wish to re-examine (2) the influence of nonlinear
models on the method of Lagrange multipliers and ocean state estimation. Is
the adjoint method useless with a highly nonlinear or chaotic system, as
studies with low-dimensional chaotic models suggest? Here we posit that the
initialization problem that has informed much of the current thinking about
the Lagrange multiplier method is not the relevant analogy for ocean state
estimation. As has been documented in textbooks

Rather than developing a new state-of-the-art data assimilation technique, we proceed by taking the existing Lagrange multipler method and developing diagnostics regarding when and why it succeeds or fails, as evaluated by the ability to fit observations. Relative to the initialization problem, the prospects for a successful state estimate are shown to be improved in the boundary control problem, even if one uses a highly nonlinear model such as the forced, chaotic pendulum. If the chaotic nature of the model is not a roadblock, what is the relevant criterion for success with the Lagrange multiplier method? Our results with the chaotic pendulum suggest that “controllability”, defined as the ability to move from one arbitrary state to another by adjustments on the control variables (e.g., external forcing), is the relevant diagnostic. The control variables of the pendulum are analogous to adjustments of the atmospheric boundary forcing in an ocean model. Therefore, there is a wide variety of situations where the Lagrange multipiers of an ocean general circulation model (GCM) are useful, and that previous GCM results can be explained in this context.

The fixed, single pendulum can be modeled as a nonlinear or linear set of
equations, and it can also be easily modified to be stable or unstable. In
many ways, the pendulum is a more flexible and easily interpreted physical
system than the often-used

We consider an “identical twin” experiment where the true solution is known
(solid line, Fig.

The rapid divergence of pendulum trajectories is indicated by the
path density of trajectories (background shading), and the evolution of
three sample trajectories: the “truth” or reference trajectory (solid line), a
“first-guess” trajectory with incorrect initial angular velocity that
diverges within 5 s (dashed line), and a first-guess trajectory with
incorrect initial angle,

We proceed by defining a least-squares cost function to be minimized. The
data-based contribution to the cost function,

A second contribution to the cost function includes two terms that constrain
the difference between our posterior and prior estimates of the initial
conditions and forcing,

Minimizing

For a nonlinear system, and a chaotic system in particular, this first-guess
trajectory usually diverges from the already-collected observations at some
point, and thus can be ruled out as a possible solution

The aforementioned standard approach does not use the observational
information already in hand that could inform the time evolution of the
forcing. There are many methods that are available to update the forcing,
such as the Kalman filter

Here we seek an update to the initial conditions and the forcing (i.e.,

The observation,

We obtain the sensitivity of

Extending the partial derivative of

In summary, the method of Lagrange multipliers is implemented with the
following steps. Starting with a guess for the initial conditions and
forcing,

Cost-function values as a function of initial angle,

Synthetic observations of the pendulum angle are generated every 2.5 s
over a 50 s time interval, where a random error of 0.5 rad is added to
every observation. We first illustrate the futility of a brute force search
for the optimal initial conditions by re-running the forward model with
combinations of the initial angle and angular velocity in the neighborhood of
the truth. The data contribution to the cost function
(Eq.

Control of the chaotic pendulum with an improved-first-guess and the
Lagrange multiplier method.

We start the application of the methods of Sect.

The improved first-guess trajectory is a better fit to the data in large part
due to the updated initial angular velocity (middle panel,
Fig.

For

Comparison of the reconstructed pendulum and truth.

State estimation of the chaotic pendulum with a reduced set of
two observations with standard error 0.5 rad

The improved first-guess trajectory better fits the observations than a
standard first-guess, but there are tradeoffs in the estimated forcing
(bottom panel, Fig.

Our identical twin experiment permits a comparison with the truth to diagnose
actual errors even at times without observations. While the improved first
guess appeared to fit the data well, the misfit to the truth displays
considerable structure, including a large deviation around

For the angular velocity and forcing (middle and bottom panels,
Fig.

In the case where only two observations are available (

In the case that many (

The previous section addresses cases where the forcing is adjusted at every
time step, leading to 5002 control variables. After application of the
Lagrange multiplier method, the resulting value of the cost function
(Eq.

For some cases where only 5 or 10 observations are available, the cost
function is small enough that overfitting may be occurring. In these cases,
we find that the control perturbations necessary to fit the data are very
small, and this impacts the size of the cost function through the

Influence of the number of observations and controls on the ability
to track the chaotic pendulum. The base-10 logarithm of the cost function,

Escaping an apparent local minimum.

We investigate the effect of a decrease in the number of controls by redefining
the external forcing control perturbation. For

For a given number of observations, a decrease in the number of controls
leads to a decrease in the likelihood of a successful fit to the data. The
initialization problem is equivalent to the case with two control variables,
and Fig.

We also find cases where the gradient-descent method is capable of navigating
the complex cost-function topology with Lagrange multiplier sensitivity
information. A slice of

Control of the chaotic pendulum with an inaccurate first guess of
the external forcing.

The previous examples in Sect. 3 proceed with prior information that the
forcing is periodic with an accurate magnitude and phase. A good analogy is
the regular forcing of solar insolation on the ocean surface. Here, we test
the performance of the Lagrange multiplier method with inaccurate prior
information about the forcing, as is a more realistic analogy to the
uncertainty of air–sea fluxes. In particular, our first guess of the forcing,

Our suggestion that controllability is a key criterion brings our
understanding of the Lagrange multiplier method into closer consistency with
the Kalman filter/smoother (i.e., the combined usage of the Kalman filter and
smoother). Both methods solve the same least-squares problem, and the
solution of a linear problem should not depend upon the chosen method

To recover the true trajectory of a system, observability is also important,
as the estimation problem is the dual of the control problem

Related to the idea of observability,

In this section, we compare criteria for the success of the Lagrange
multiplier method. Previously suggested criteria include Lyapunov exponents
or other stability metrics of the tangent-linear model

Here we investigate the influence of stability versus that of nonlinearity.
The pendulum is a useful system because it is easily modified to have
four distinct dynamical states: (1) nonlinear, unstable, (2) nonlinear, stable,
(3) linear, stable, and (4) linear, unstable. Case (1) is the original dynamical
equation for the pendulum (Eq.

We revisit the problem of estimating the initial angle when

Cost function with respect to the initial pendulum angle. A
synthetic observation was made from a model run with initial angle,

Conversely, the linear, unstable case does yield a parabolic cost function
(Fig.

In the Introduction, we remarked on the only ocean state estimate known to
the authors that successfully implemented the Lagrange multiplier method in
an eddy-permitting ocean GCM without any modification to the adjoint model

Nested view of the

Nonlinearity is not a fundamental obstacle to constraining a model to observations using the Lagrange multiplier method. On the basis of research primarily with toy models, chaotic systems were thought to represent such an obstacle if the estimation time window was too long. Here we find that the trajectories of the nonlinear pendulum can be tracked over multiple rapid transitions that are due to chaotic dynamics. The Lagrange multiplier method is successful under the condition that enough boundary controls are available through time, and that the system passes a test of controllability. In the case of the pendulum, the rank of the controllability matrix is a better metric to predict a success of state estimation rather than a measure of dynamical stability. The ocean state estimation problem is analogous to the problem posed here; uncertain air–sea fluxes contain large errors that require control adjustments through time.

Our implementation of the Lagrange multiplier method includes a step to
construct a good first guess that helps the iterative gradient-descent
search. The first-guess method has been developed with implementation in an
ocean GCM in mind. Specifically, sub-problems are defined over the interval
between observations and thus require less memory than a whole-domain
approach. In addition, we suggest that the particular first-guess method of
this work is not the only way to produce a good first guess, and that other
methods would bring the first-guess state close enough to the truth to
increase the likelihood of success. A good example is the Green's function method

No datasets were used or produced in this work.

The forced, nonlinear pendulum is governed by the following equation,

The linear, stable pendulum is derived with the small-angle approximation.
This approximation is a linearization around zero displacement

The authors declare that they have no conflict of interest.

We thank Geir Evensen, Armin Köhl, Olivier Marchal, and Eli Tziperman for discussions on this topic over the last decade, and to Jacques Verron for his note that has encouraged this work. Geoffrey Gebbie also acknowledges Carl Wunsch, Patrick Heimbach, Detlef Stammer, and Julio Sheinbaum for guidance getting started on this project. Geoffrey Gebbie was funded through the Ocean and Climate Change Institute of the Woods Hole Oceanographic Institution. Tsung-Lin Hsieh was funded by the Arthur Vining Davis Foundations Fund for Summer Student Fellows through the Woods Hole Oceanographic Institution. Edited by: Zoltan Toth Reviewed by: three anonymous referees