In a broad spectrum of scientific fields, transferring the information contained in L observed data time series yl(tn); l = 1, …, L; n = 0, …, m to a physically based model of the processes producing those observations allows the estimation of unknown parameters and unobserved states of the model within an observation window {t0,…,tm}. As a sample of these fields, we note applications in meteorology , geochemistry , fluid dynamics and plasma physics , among many others.

The conditional probability distribution P(X) = exp⁡[-A0(X)] allows evaluation of expected values of physically interesting functions G(X) along the path. P(X) is conditioned on the measurements yl(tn); l = 1, …, L; n = 0, …, m. These are L-dimensional, while the model state is D-dimensional; it is usually the case that D ≫ L. Data assimilation seeks to use the information in y(tn) to estimate unknown parameters in the model and unobserved states of the model. If this is accomplished, one uses prediction for t > tm to examine the validity of the model.

The action A0(X) contains terms giving a measurement's influence on P(X), terms propagating the model between the measurement times tn, and a term -log⁡[P(x(0))] representing the uncertainty in the initial state . We discuss the familiar case where additive measurement errors are independent at each tn and Gaussian with covariance Rm-1(l, l′, t); l, l′ = 1, …, L. The model is a physical differential equation, discretized in space and time and satisfying the D-dimensional stochastic discrete time map xa(n + 1) = fa(x(n)) + Rf-1/2(a, b)ηb(n); a, b = 1, …, D with iid Gaussian noise error ηa(n) ∼ N(0, 1). We take Rm(l, l′, n) = Rm(n)δl,l′ and Rf(a, b) = Rfδa,b. Rm(n) is zero except near observation times tn.

With these conditions, the action takes a familiar form: A0(X)=∑n=0mRm(n)2∑l=1Lxl(n)-yl(n)2+Rf2∑n=0m-1∑a=1Dxa(n+1)-fa(x(n))2-log⁡[P(x(0))].

The conditional expected value 〈G(X)〉 of a function G(X) on the path X = {x(t0),…,x(tm)} is given as 〈G(X)〉=∫dXG(X)exp⁡-A0(X)∫dXexp⁡-A0(X).

One interesting function G(X) is X itself, whose expected value gives us the average path over the measurement window [t0, tm]. Estimates of the parameters and P(x(tm)) permit prediction for t > tm; in this window, one compares observations and predictions using model output with a user-selected metric. No information is passed to the model in the prediction window. Importantly, good estimation of observed state variables (a “good fit”) is not sufficient to produce confidence in the quality of the model; good prediction is critical.

To approximate the integral 〈G(X)〉, we follow the stationary path method of Laplace and seek saddle points in path space Xq labeled by q = 0, 1 … satisfying ∂A0(X)/∂Xα = 0. The index α is a label for the state and time α = {a,n}. To determine their importance in evaluating the integral, Eq. () paths are sorted by increasing action levels A0(Xq): A0(X0) ≤ A0(X1) ≤ ⋯.

We take the the usual data assimilation technique several steps further by

showing how to find a consistent path X0 for the minimum action level using an annealing method,

For nonlinear problems of interest, there may be many paths Xq; q = 0, 1, … satisfying the saddle point condition. To assess their contributions to 〈G(X)〉, we expand A0(X) in the neighborhood of each Xq as (note that all variables are notated in Table ) A0(X)=A0Xq+X-Xqα1γα1α22XqX-Xqα2+∑r=3A(r)Xqα1…αrr!X-Xqα1…X-Xqαr. There is an implied sum over all paired Greek indices αj. A sum over all terms with comparable action level A0(Xq) then gives an approximation to 〈G(X)〉.

Changing variables to Uα = γαβ(Xq) (X - Xq)β leads to a contribution to the numerator of 〈G(X)〉 in Eq. () from each Xq exp⁡-A0XqdetγXq∫dUexp⁡-U2-VXqGXq+WXq where VXq=∑r=3A(r)Xqα1…αrr!γXq-1Uα1…γXq-1Uαr, and WXq=∑k=1G(k)Xqα1…αkk!γXq-1Uα1…γXq-1Uαk. The contributions to the denominator of Eq. () are identical to Eq. (), with the factor [G(Xq) + W(Xq)] replaced by unity. We sum over the contribution of each Xq to evaluate 〈G(X)〉.

If the lowest action level A0(X0) is much smaller than all others, then exp⁡[-A0(X0)] ≫ exp⁡[-A0(Xq≠0)] and its contribution to 〈G(X)〉 totally dominates the integral. We have then that 〈G(X)〉=∫dUexp⁡-U2-VX0GX0+WX0∫dUexp⁡-U2-VX0, plus exponentially small corrections from the action levels associated with other saddle paths Xq≠0.

We now turn to an annealing method to find the path X0 where exp⁡[-A0(X0)] ≫ exp⁡[-A0(Xq≠0)]. Within this method, we first examine the importance of the number L of measurements at each observation time tn. We then present an argument that, in the integral Eq. (), contributions to 〈G(X)〉 from the terms V(Xq), W(Xq) behave as inverse powers of Rf as Rf/Rm becomes large. This would leave us with 〈G(X)〉 ≈ G(X0) with corrections that are a power series in Rf-1. The implication is that all statistical data assimilation questions, as Rf/Rm becomes large, can be well approximated by the contribution of the path X0 with the lowest action level A0(X0) along with corrections one can evaluate via standard perturbation theory. Variations about 〈G(X0)〉 would be small and computable.

The term in the action expressing uncertainty in the initial model state x(0), P(x(0)), is often written assuming Gaussian variation about some base state xbase, so -log⁡[P(x(0)] ∝ (x(0) - xbase)2Rbase/2, and this has the form of the measurement term evaluated at n = 0. We incorporate this expression into the term with coefficient Rm in the action and no longer display it. This term, often called a prior, is also seen at the initial condition for the time evolution of the conditional probability distribution P(X).

We now present the results of a set of calculations on the Lorenz96 model used frequently in geophysical data assimilation discussions as a test bed for proposed methods. The model has D degrees of freedom xa(t) satisfying the differential equation x˙a(t)=xa-1(t)xa+1(t)-xa-2(t)-xa(t)+f+ζa(t); a = 1, …, D; x-1(t) = xD-1(t), x0(t) = xD(t), xD+1(t) = x1(t); ζa(t) is N(0, Rf-1) Gaussian noise. f = 8.17, for which the xa(t) are chaotic . We studied D = 20 and added f as an additional degree of freedom satisfying f˙ = 0. The number of model equations in the action is therefore equal to D + 1.

We performed a twin experiment in which we solved these equations, with ζa(t) = 0 and with an arbitrary choice of initial conditions x(0) using a fourth-order Runge–Kutta solver with Δt = 0.025 over 160 steps in time. Here, t0 = 0 and tm = T = 4. We then added iid Gaussian noise with zero mean and variance σ2 = 0.25 to each time series. L = 1, 2, … of the data series were then represented in the action at each measurement time tn during our annealing procedure. In our calculations, Rf0 = 0.01 and ξ = 2.

In the action, we selected Rm = 4, the inverse variance of the noise added to the data in our twin experiment, so the minimum action level we expect is 80.5 L ± 8.97 L. The paths are (m + 1) (D + 1) = 3381-dimensional. Our search for minimum paths used a BFGS routine to which we provided an analytical form of the gradient of A0(X). The search was initialized with NI = 100 times with initial paths from a uniform distribution of values in the interval [-10, 10].

We also investigated using the IPOPT public domain numerical optimization package and found that it also worked well, giving essentially the same results as BFGS. Any existing method for 4DVar should work as well for this annealing procedure.

In Fig. , we display the computed action levels for L = 5, 7, 8 and 9 at each value of log⁡Rf ∝ β. For L = 5, there are many close action levels associated with the extremum paths of the action Eq. (); as L increases, the lowest action level visibly separates from the others. At the bottom of each panel, we indicate the lowest action level value and its standard deviation. The next-lowest action level A0(X1), for L = 5, 7, 8, and 9, is at 403.8, 749.8, 1161.6, and 2256.1, respectively. This means that for L = 5 we would have to sum over the contributions of many paths Xq to evaluate the expected value 〈G(X)〉. At L = 7 or higher, X0 dominates the integral. Our estimate for the forcing parameter, set to 8.17, was 8.22 at large β.

We think it important to note that if we had begun our search for the saddle point paths Xq at large values of Rf, we would be almost sure to miss the actual path X0, which gives the lowest action level, since the Hessian matrix of A0 is ill conditioned when Rf is large; see Fig. 4.6 in .

Another sense of why beginning a search at large values of Rf may fail to find the action level identified in the annealing approach is that the basin of attraction of the minimum action level is likely to be so small, relative to the size of path space, that “falling into it” through an arbitrary initial path used in any version of a variational procedure is unlikely. The annealing method systematically tracks the known minimum for very small Rf and, in that manner, starts in and remains in the appropriate basin of attraction.

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 t > tm = T .

As this is a twin experiment, we show in Fig.  the data, the estimated state variable, and the predicted state variable for an observed variable x3(t) and for an unobserved variable x12(t) for L = 8. In a real experiment, we could not compare our estimates for the parameters or the unobserved state variables. Although the estimation procedure for the path X0 with the minimum action value is rather good, both in estimating the 12 unobserved states and the one parameter, there nevertheless exist errors in our knowledge of the full state (x(t) = 4). The predictions thus lose their accuracy in time because of the chaotic nature of the trajectories at f = 8.17.

In the Lorenz96 equations, one usually has a single forcing parameter f. To see how well our procedure works for several unknown parameters, we introduced ten different forcing parameters fa into the Lorenz96 model at D = 10: x˙a(t) = xa-1(t) (xa+1(t) - xa-2(t)) - xa(t) + fa, that is, with no fluctuations in the dynamics. Noise was added to the solutions to the Lorenz96 system before the xa(t) are used as data. For D = 10, the lowest action level stands out from the rest at L = 4. In Table , we show our estimates for the ten forcing parameters for L = 4, 5, and 6, as well as the actual value used in the calculations of the data. In these estimates, and for the single forcing parameter reported above for D = 20, there is a known source of bias . As one can see in the examples, it is small here.

In the twin experiments we presented noisy data from the known model as L = 1, 2, … data series of observations. To give some sense of what one might expect if the model were totally wrong, in the sense that the data we presented came either from a completely different model system or from enormously noisy data, we presented data from a collection of 1963 Lorenz model oscillators to a Lorenz96 D = 10 model. Twelve time series data are generated by four individual Lorenz63 systems with different initial conditions. Gaussian white noise with zero mean and standard deviation σ = 0.5 are added to each time series. All these “data” yl(t) are rescaled to lie within [-10, 10].

We then place these signals as “data” in the action with the model taken as Lorenz96 D = 10; the single forcing parameter is treated as a time-dependent state variable obeying f˙ = 0. We use L = 5 and L = 6 as measurements using the data time series taken in the order y1(t), y3(t), y5(t), y7(t), y9(t) for L = 5, and y1(t), y3(t), y5(t), y7(t), y9(t), y2(t) for L = 6. In Fig.  we display the action levels versus log⁡[Rf/Rf0] = β associated with this for L = 5 and L = 6. Results for other values of L are consistent with these. This example provides a graphic illustration of the inconsistency of the data and the model and how this makes its appearance in the annealing procedure.

We also investigated, but do not display here, the action levels when the parameter f is held at a different value in the model than was used for generating the data. In particular, we generated the data with f = 8.17 and then held f = 18 in evaluating the action levels for a Lorenz96 model with D = 5 and L = 1 and 2. In each case, no minimum action level split from the collection of Xq and no level was close to the consistent χ2 condition discussed above. This actually emphasizes the importance of allowing the variational method to include all parameters as state variables with a zero vector field, allowing the estimation of the parameters along with the unmeasured state variables.

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, L, has been demonstrated. We have no formal proof that the path X0 gives a global minimum of the action; our criteria of consistency with Eq. () and excellent predictions after the observation window are functionally useful features of the procedure.

We turn back to the evaluation of the path integral for 〈G(X)〉 in Eq. (). In that integral for our action, we have 12A0(2)(X)=γαβ2(X)=Rm(n)δa,lδb,l′+Rfhαβ(X)andA0(r)(X)=Rfg(r-2)(X), for r ≥ 3. The functions h(X) and g(X) are derivable from the form of A0(X). These are to be evaluated at X = Xq for the qth saddle path.

In the form of the integral Eq. (), we see that the term in the exponential with order r in U has an order of magnitude about Rf/(Rm + Rfh(Xq))r/2 for r ≥ 3. Similarly, in the expansion of G(X), the term of order k has a denominator of order 1/(Rm + Rfh(Xq))k/2 for k ≥ 1. Since only terms with even powers of U are nonzero in the integral because of symmetry, we have a collection of terms that, as Rf becomes large with respect to Rm, decrease as 1/Rf or faster. As a rule of thumb in the calculations we presented for Lorenz96 D = 20, we see that for β = 15 or larger, or Rf/Rm ≈ 100 or larger, the terms in the path integral beyond the quadratic term in the expansion of the action become small. A stronger estimate would come from evaluating the leading term in 1/Rf.

We have examined the path integral formulation of data assimilation Eq. () and asked how well a variational approach to the conditional expected values of functions G(X) on the path X = {x(0),x(1),x(2),…,x(m)} can approximate the integral. We use Laplace's method, which estimates the path integral by seeking saddle paths of the action where ∂A0(X)/∂Xα = 0. This approximation is widely used in meteorology and other fields, where it is known as weak 4DVar . The action is the cost function that is minimized.

When measurement errors and model errors are distributed as iid Gaussian noise, we have described an annealing method in the strength Rf with which the model errors enter the action that permits the following of a collection of action levels from Rf = 0, where the saddle paths are known precisely, through a set of increasing values of Rf at each of which a numerical optimization algorithm produces a set of saddle paths that are used as the initial conditions for the solution of the variational approximation at the next larger value of Rf. This permits the identification of a lowest action level associated with a saddle path X0 where A0(X0) is split from the action values for the other possible saddle paths Xq>0) at each Rf: A0(X0) < A0(Xq>0)). The contribution of X0 then dominates the integral.

We also explored the dependence of the action levels revealed through the annealing method on the number of measurements L made at each observation time. When L reaches and exceeds a critical value, related to the number of unstable directions in the deterministic chaotic dynamics of the model, the contribution of the path X0 is either certainly the dominant contribution to the conditional expected value of G(X), or it is the only path satisfying the saddle path condition. By expanding the integrand of Eq. () about X0 we argued that the resulting corrections to the contribution of X0 produced a power series in Rf-1.

In previous work with variational principles for data assimilation, we are unaware of any procedure such as our annealing method using Rf to identify a consistent minimum action. Nor are we aware of a systematic exploration of the dependence of the action levels on the number of measurements at each observation time. Of course, there is the discussion in which suggested this work. Finally, we do not know of any previous discussion of the corrections to the variational approximation, which here is shown to consist of small perturbations when the resolution of the model error term in the action is increased; namely, Rf becomes large.

The relation of the annealing method to familiar 4DVar calculations is simple to state: each set of calculations during the annealing in Rf, at each value of Rf, is a 4DVar calculation. The annealing has the added advantage of allowing one to establish an identified lowest action value path X0 that gives the dominant contribution to the quantities of interest, namely, the conditional expected values of functions G(X) on the path.

A similar annealing procedure in the importance of the model errors as represented by Rf can be incorporated into a Monte Carlo calculation of the path integral. The relation between that method of approximating the high-dimensional integral and the variational method we have focused on here is through the Langevin equation in “algorithmic time” described in . In each approach, the statistics of moments about X0 or marginal distributions of selected state variables is achieved by the choice of G(X). This paper has compared those evaluations in the variational method to standard 4DVar by using 4DVar at each stage of the annealing adding the identification of the path X0 with the lowest action level. By performing the annealing procedure in a Monte Carlo context, one could compare it to other standard ensemble methods such as the ensemble Kalman filter (EnKF) . However, that is a subject for another investigation of the annealing approach; we have focused here only on variational methods.

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 Rf. We have shown that the path that gives a consistent minimum action level can be traced by an annealing procedure starting with a setting where the dynamical model essentially plays no role, Rf ≈ 0, then systematically increasing the influence of the model dynamics. As the scale of the model error reaches about 100 times the scale of the measurement error, the path X0 associated with a consistent global minimum action level dominates the path integral with corrections of order 1/Rf. The important role of the number of measurements L at each measurement time is also demonstrated.

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 X0 giving that lowest action level is a power series in Rf-1 may not succeed. The precise way Rf enters the matrix γαα′2(X0) determines that.

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 A0(X0) to lie well below the next level. For a solution to that problem, we have used information from the waveform of the measurements as shown in some detail in .

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. (). We anticipate that our use of Gaussian model error and measurement error is a convenience and that other distributions of these errors will permit many of the same set of statements about the value of variational approximations to statistical data assimilation.