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Edited by: Wilhelm Stannat, Technische Universität Berlin, Germany

Reviewed by: Xin Tong, National University of Singapore, Singapore; Meysam Hashemi, INSERM U1106 Institut de Neurosciences des Systèmes, France

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

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The goal of this work is to analyse and study an _{0}, _{n}] under consideration. Then, an

Data assimilation is concerned with the use of observation data to control or determine the state of some dynamical system [

Over time several generations of data assimilation methods have been developed, for example optimal interpolation in the 70th, variational methods in the 80 and 90th and ensemble data assimilation since about 1995, with very intense research activities since about 2000 (e.g.,[

Usually, data assimilation takes ^{(b)} (also called ^{(a)}. Usually, the estimation of the analysis state is performed in turns with short-range forecasting, i.e., within a given temporal framework assimilations are carried out at times _{k} for _{k−1}. Then, the core analysis step is carried out at time _{k}, based on observations which are available either at _{k} or in the interval [_{k−1}, _{k}]. Alternating short-range forecasts and core analysis steps leads to the classical

Today, many forecasting systems have moved away from pure deterministic forecasting and employ

Often, for large-scale realistic systems, the model forecast as well as the analysis step needs huge computational resources. They limit the temporal resolution of the data assimilation cycle. Further restrictions are given by the availability of observations, which need to be measured and distributed to reach operational centers. For example, to run an assimilation cycle of 1 h for convection permitting high-resolution numerical weather models, top-500 supercomputers are needed to achieve a sufficient resolution and spatial extension of the model fields under consideration [

The core task addressed in this work is the problem of _{0}, _{N}]. The next classical analysis is calculated for time _{N}, such that a similar ensemble forecast will be available for a subsequent interval [_{N}, _{N+1}]. Here, we limit our interest in the ultra-rapid data assimilation for observations _{k} which are available at points in time _{k} with _{0} < _{1} < _{2} < … ≤ _{N}. The task is to provide an update of the ensemble forecast with high speed without using the full numerical model or a full-grown data assimilation system. In particular when we are interested only in the forecast of some layer or part of the state space, this is of high practical interest.

Usually, the classical forecast cycle in operational centers is based on a data assimilation cycle with frequency _{N} of several hours. The term

We will base ultra-rapid data assimilation on the ensemble transformation matrix given by the ensemble Kalman filter (EnKF) or ensemble Kalman square root filter (SRF), compare [^{1}

To study the quality of ultra-rapid data assimilation we apply the basic ideas to the Lorenz 1963 model [[_{k}, _{0}, _{N}].

The approach discussed here was first suggested in the work of Etherton [

In section 2 we introduce our notation and basic results from ensemble data assimilation. In particular, we introduce the ensemble Kalman filter in the notation of Hunt et al. [

This section serves to collect notation and basic results on the ensemble Kalman square root filter (SRF) following the notation of Hunt et al. [

We consider a state space ℝ^{n}, an observation space ℝ^{m} with ^{n} and observations ^{m}. The basic idea of the ensemble Kalman filter type methods such as the SRF is to approximate the covariance matrix ^{n×n} of the system based on some ensemble ^{b, (ℓ)}, ℓ = 1, …,

where

is the matrix ^{b} ∈ ℝ^{n×L} of centered differences (sometimes its columns are called the

Note, by construction the space spanned by the member of the centered ensemble has dimension

In order to assimilate observation data the ^{b, (ℓ)} of the ensemble member are required, which are obtained by applying the observation operator ^{n} → ℝ^{m} to the corresponding ensemble member

With these quantities the matrix ^{b} ∈ ℝ^{m×L} can be defined analogously to ^{b}

which one also denotes as ^{b} = ^{b} assuming a linear operator

In the following, + between a vector and a matrix indicates a column-wise summation _{1}, …, _{L}) with the columns _{ ℓ }, ℓ = 1, …,

with the transformation matrices ^{m×m}. We quickly review the different versions as follows. An analysis update of the centered ensemble (see Equation 4) given by

leading to Equation(8). In Equation (9) we have used an update of the ensemble mean

with ^{b}. The definition of the transformation matrix

leads to the update Equation (11). For the full transform matrix _{full} we obtain

based on

and by

Different notations have been used over time, depending on whether you want to keep your equations close to the classical Kalman filter equations or for a more practical focus. The quantities defined in Equations (2, 6) differ from the definitions of ^{b} and ^{b} of Hunt et al., c.f. Equations (12, 18), by the normalization factors. The relations are

The full ensemble matrix has different letters ^{a} =

the update of the mean Equation (14) rewrites as

with

The transformation matrix in the sense of Equation (25), giving us the increment in ensemble space, is now given by

The ensemble Kalman filter combines the above introduced notion of an ensemble of model states to describe spatial and temporal correlations with the well-known Kalman Filter [

with the Kalman gain matrix

and the transformation matrix

Taking the square root of the symmetric matrix

which is the transformation matrix of the update for the centered ensemble in Equation (8). Note, the notation in Equations (2, 6) is the one used by Nakamura and Potthast [^{a} defined by Hunt et al, the identity ^{a} =

The update of the mean is obtained along the lines of the classical Kalman filter by

with K given in Equation (28). Comparing Equation (14, 31) leads to

in case of the SRF. The update for the full ensemble using the ensemble Kalman square root filter is therefore given by applying Equations (30, 31) to Equation (11, 15).

Here, we can now confirm the validity of Equation (17). From the definition of ^{b} we know that the sum of the rows of ^{n×L} is zero, and the sum of the columns of ^{b})^{T}^{b})^{T} ^{b})^{T} ^{b})^{T}

Before we investigate ultra-rapid data assimilation based on reduced data, we need to recall how a standard ensemble Kalman square root filter will react when we base our analysis on a reduced set of model variables. Let us study the calculation of the ensemble analysis for the ensemble Kalman filter with reduced data. The basic formula for the ensemble Kalman filter can be expressed as

Now, assume we observe ^{m} which depends on some subset _{1}, …, _{ñ} of the full set of variables _{1}, …, _{n} only. Given these reduced spaces the operator

In this case, the terms ^{b} ∈ ℝ^{m×L} and (^{b})^{T} ∈ ℝ^{L×m} will be a linear combination of the variables 1, …, ñ of the ensemble members. If we are given the variables _{1}, …, _{ñ} of the ensemble members only, the matrix ^{b} will depend only on the variables ^{b}. The solution ^{m} of

is calculated based on the variables _{1}, …, _{ñ} of ^{b} and ^{b} only. We summarize the result of these arguments in the following lemma.

L_{1}, …, _{ñ}^{n} of the state space of our dynamical system, the transformation matrix W of the ensemble Kalman square root filter update x^{a} − x^{b} depends on these variables of the centered ensemble Q and the mean first guess

A consequence of the above Lemma 0.0.1 is that, if we have reduced observations, the ensemble Kalman square root filter will give us an update matrix

But we need to pay attention to the update and propagation step. The update Equation (11) clearly updates all variables, since ^{L×L}, and thus all variables of _{1}, …, _{ñ}.

Clearly, in general we cannot run the full ensemble Kalman filter on a reduced set of variables, just because you need all prognostic variables to run the numerical model. We will see later, that this limitation does no longer apply when we are in the framework of ultra-rapid data assimilation.

This section serves to develop the main ideas of ultra-rapid _{k} are given at point of time _{k}, _{0}, _{N}] for which we are not able to employ a full data assimilation functionality. We assume that we have been able to perform some ensemble data assimilation scheme prior to the time _{1} at time _{0} and that a forecast ensemble has been calculated, such that

is available at the _{ξ}, ξ = 0, …, _{1}, _{2}, …receive observations _{1}, _{2}, … The goal is to provide ultra-rapid updates for estimation of our state at times _{1}, _{2}, … When we are at time _{k}, we would like to update the forecasts at the times _{ξ} for ξ =

Note, the assimilation of observations at some point in time exhibits information about the past as well. This is called

Assume that we are given some ensemble _{k} ∈ ℝ, which could be an analysis or a first guess from somewhere. Further, we assume that we have applied our model _{k+1}, …, _{N} for

We employ the following matrix notation. The matrix

of forecasts _{k} to _{ξ}. The matrix ^{(k)} is the matrix of linear ensemble transform coefficients calculated based on the observations _{k} at time _{k} and the first guess ensemble at time _{k}, i.e.,

When the analysis ensemble at time _{k} is given by a generic ensemble data assimilation approach, we know that

with the matrix _{k} for which _{j, ℓ} is calculated. Also, we note that the background

L_{ξ} when observations at time k are assimilated by a linear data assimilation method as in Equation (12), the forecast ensemble can be calculated by

for ℓ = 1, …, _{k − 1, ξ} = _{k, ξ}_{k − 1, k}. □

Before we continue with our introduction of ultra-rapid data assimilation, we would like to study the reduced variable case in the above Lemma 0.0.2. Clearly, to apply _{k, ξ} to a state ^{(a)} or ^{(b)}, we need to know the full state. If only a part of the state _{k − 1} to time _{ξ}.

C_{1}, …, x_{ñ} of the state x only, then the transformation matrix W^{(k)} for the assimilation of y_{k} can be calculated from a) the first guess ensemble data_{k}. For a linear model M, for the variables with index i we have

_{k} can be calculated from the knowledge of the reduced set of variables only

The consequence of Equation (41) is that for linear models we can calculate the forecast based on the analysis at time _{k} by a superposition of the forecast from time _{k − 1}. The weight matrix _{k} given by the linear ensemble data assimilation scheme. We can also use Equation (41) recursively, which is formulated in the following Theorem.

T_{j}, j = 1, …, k at times t_{1}, …, t_{k}. The goal is to calculate the forecasts_{ξ} based on the observations from t_{1} to t_{k} and the initial ensemble_{0} with an ensemble data assimilation method as in Equation (11). If the model M is linear, we obtain

for ξ =

and by the same step

for

Note that the recursive application of Equation (41) implies that any transformation matrix ^{(i)} is obtained using the observation _{i} and the full ensemble _{i − 1, ξ}.

The results for reduced data are also valid for the core formula (43). We collect the relevant statements into the following corollary. The matrix _{k, ξ} contains the different state variables in its rows and the columns represent the ensemble under consideration. We employ the notation (_{Fk, ξ)i = 1, …, ñ} for the rows with the variable indices

C_{1}, …, x_{ñ} of the state x only, then the transformation matrix ^{(k)} _{k} can be calculated from a) the first guess ensemble data_{0, k})i _{k} and c) the previous transformation matrices^{(1)}⋯^{(k − 1)} _{1}⋯y_{k − 1}. For a linear model M, for the variables with index i we have

i.e., the formula (43) is valid and the ensemble forecast based on the analysis with observation _{k} can be calculated from the knowledge of the reduced set of variables only.

Smoothers are schemes which employ information from the future to improve the estimate about some present state. Alternatively, you could say that they use information now to update past states.

When we consider the scenario of ultra-rapid data assimilation, for the interval [_{0}, _{N}] we are given an ensemble of original states (35) over the full interval. When an observation is arriving at time _{k} (ignoring delay usually needed for observation processing and transfer), we can employ the same techniques which are used for updating the analysis and forecast to the past interval [_{0}, _{k}].

D

Given the original first guess ensemble _{0, ξ} for ξ = 0, …, _{0}, _{N}] we define the ensemble analysis given the data _{1}, …, _{k} by

This analysis ensemble is defined for the full time interval.

In general, a convergence analysis of an ensemble Kalman smoother and its comparison to a four-dimensional variational data assimilation (4D-VAR) scheme over the time window [_{0}, _{n}] can be found in Theorem 5.4.7 of Nakamura and Potthast [

Clearly, if we replace the full model _{true}, the temporal correlations, which are implicitly used when we employ the analysis matrix ^{(k)} to update the ensemble in the past or in the future, may not be correct with respect to the true ensemble correlations. In this case, the information _{k} in the future of _{0} may not improve the state estimate at time _{0}, but lead to additional errors in this state estimate. We will demonstrate this phenomenon in our numerical examples in section 4.

The goal of this section is to study the ultra rapid data assimilation for simple generic examples. We want to show that the assimilation step can be carried out in a stable way and that the ultra-rapid forecasts indeed show an advantage over the ensemble forecasts without this step. Also, we would like to understand the range of skill which we can achieve when we compare it with the full standard data assimilation and forecasting approaches.

Here, we start our study with the Lorenz 63 model Lorenz [

The Lorenz 1963 model is a system of three non-linear ordinary differential equations

with constants

We show the simulation of some trajectory by the Lorenz model in black, the first 8 cycles in

Here, we want to test the feasibility of ultra rapid data assimilation. The original curve is shown in black in Figure _{1}, _{2}, …, _{k} with Δ_{t} = _{i+1} − _{i} = 0.1 (without units). For the original, we have used the above ODE system with

We have now followed two tracks. First, we have implemented an ensemble Kalman square root filter. We start with a first guess ensemble, which is generated at time _{0} by adding random Gaussian errors to the starting point of the original curve. Then we assimilate the observations (the black dots) using the Ensemble Kalman square root filter.

Second, the ultra-rapid data assimilation and forecasting cycle has been implemented. The ultra rapid data assimilation has been set up by first calculating the full first guess ensemble for the whole time interval under consideration. Then, a modified ensemble is calculated step by step following (43). We study ^{(k)} based on the observations _{k} at time _{k},

The result of _{k} with

We show the results of the ultra rapid data assimilation in comparison with the ensemble Kalman square root filter for the Lorenz 1963 model for

Here, we also investigate the ultra rapid data assimilation tool as a smoother. We calculate the analysis ensemble _{0}.

In Figure _{true} is different from the model

Studying the results of the ultra-rapid ensemble smoother over _{0} and _{N} for _{1}, _{4}, _{7}, …, _{3}1, starting with a thin blue curve and ending with a thick red curve.

Studying Figure _{0}, _{N}] increase when we assimilate more and more data. When the ensemble reflects the correct correlations between the future and the past, the error should decrease. However, with a numerical model which is different from the true model, we also inherit errors into the temporal correlations. As a consequence we observe that the error at _{1} increases when we assimilate further data _{k} for _{0}, _{N}].

In Figure _{0} = _{0}), which is used to obtain the truth as well as the ensemble. We evaluate differences of the corresponding mean from the truth and take appropriate ratios. In Figure

We show the results of the mean-error of the ultra rapid data assimilation in comparison with the original first guess (no data assimilated) and the ensemble Kalman square root filter for the Lorenz 1963 model. We used _{stat} = 250 different initializations of the random number generator to obtain different distributions for the observations and the initial ensemble. After assimilation of all data the mean error at each time step on the trajectory from the truth is counted. In

At the end of this section we highlight the impact of the time step in the model, which translates to the time the forecast from one point on the trajectory is performed. Note, this does not affect the performance of the Runge-Kutta-Scheme where the time step of the integration is kept fixed. We evaluate the ratio of the deviations from the mean error from the SRF to URDA. In Figure _{stat} = 250 and the total number of time steps ^{1}/_{4} of the standard time step size

We show the ratio of the mean-error of the SRF divided by the one of URDA for different step sizes in time. Specifically, in

In the second part of our numerical study we would like to understand how ultra-rapid ensemble data assimilation can be applied to the case where only a reduced set of variables is passed down from the standard ensemble data assimilation framework.

In the framework of the Lorenz model, we have carried out a study the use of the observation operators

and study assimilation of observations of either the full state

We note that _{1}, _{2} or _{3}, i.e., _{1}_{1} of _{2}_{1}, _{2}) of

Here, we focus on the results for the use of _{2} in Figure

We show the results of the ultra rapid data assimilation in comparison with the ensemble Kalman square root filter for the Lorenz 1963 model for _{2}. In

We analyse and investigate a ultra-rapid data assimilation scheme based on an ensemble square-root Kalman filter. Here, we have studied the analysis cycle, a preemptive forecasting step and also an ultra-rapid ensemble smoother.

For linear systems we have shown that the ultra-rapid data assimilation is equivalent to the full ensemble square-root filter. For non-linear systems, the Lorentz 63 system serves as a standard test case which is widely used within geophysics or the life sciences. We have carried out numerical tests of the URDA scheme, which shows highly encouraging results. For a significant number of assimilation and forecasting steps the URDA scheme shows a similar forecasting skill as the square-root filter with full model forecasts.

In particular, we have analyzed and tested the assimilation of observations which are influenced by a selection of state variables only, where the URDA scheme provides the possibility to touch only the variables of interest for the assimilation and preemptive forecasting or smoothing steps. This has very-high potential for many applications, where high-frequency analysis and/or forecasts need to be calculated, e.g., in the area of brain surgery in neuroscience or in nowcasting in geophysical applications.

This work aims to provide the basic theoretical inside and study a standard non-linear system of wide interest, the Lorenz 63 system. Initial tests on a real-world system in geophysics have been carried out in Etherton [

Ideas for the investigation by RP. Execution of the numerical calculations were performed by RP and CW. Writing the publication was done by RP and CW.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Jeffrey Anderson from NCAR for the interesting discussions and pointing us to the paper of L. Madaus and G. Hakim. Furthermore we thank Z. Paschalidi and J. W. Acevedo Valencia for fruitful discussions on the topic, applications and further development of these ideas in the context of the projects Sinfony and Flottenwetterkarte at Deutscher Wetterdienst. This work was supported by the Deutscher Wetterdienst research program Innovation Programme for applied Researches and Developments (IAFE) in course of the SINFONY project.

^{1}Note that the calculation of this transform matrix takes place in ensemble space and is only a very small part of the total cost of the assimilation cycle and forecasting.