13.7. Calculation algorithm “EnsembleKalmanFilter

13.7.1. Description

This algorithm realizes an estimation of the state of a dynamic system by a Ensemble Kalman Filter (EnKF), avoiding to have to perform the tangent or adjoint operators for the observation and evolution operators, as in the simple or extended Kalman filters.

It applies to non-linear observation and incremental evolution (process) operators with excellent robustness and performance qualities. It can be interpreted as an order reduction of the classical Kalman filter, with a remarkable assimilation quality of this filtering for large problems. It can be compared to the Calculation algorithm “UnscentedKalmanFilter”, whose qualities are similar for non-linear systems.

We notice that there is no analysis performed at the initial time step (numbered 0 in the time indexing) because there is no forecast at this time (the background is stored as a pseudo analysis at the initial time step). If the observations are provided in series by the user, the first one is therefore not used. For a good understanding of time management, please refer to the Timeline of steps for data assimilation operators in dynamics and the explanations in the section Going further in data assimilation for dynamics.

In case of linear of “slightly” non-linear operators, one can easily use the Calculation algorithm “ExtendedKalmanFilter” or even the Calculation algorithm “KalmanFilter”, which are often far less expensive to evaluate on small systems. One can verify the linearity of the operators with the help of the Checking algorithm “LinearityTest”.

There are many deterministic or stochastic variants of this algorithm, allowing in particular to perform size reduction of algebraic problems at different levels (by using reduced rank methods, dimension reduction, changes of computational space, leading to schemes of type Ensemble Square Root Kalman Filters (EnSRKF) or Reduced-Rank Square Root Filters (RRSQRT), to deterministic transformations…). We do not go into the complex details of classifications and algorithmic equivalences, which are available in the literature. The following stable and robust formulations are proposed here:

  • “EnKF” (Ensemble Kalman Filter, see [Evensen94]), original stochastic algorithm, allowing consistent treatment of non-linear evolution operator,

  • “ETKF” (Ensemble-Transform Kalman Filter), deterministic EnKF algorithm, allowing treatment of non-linear evolution operator with a lot less members (one recommends to use a number of members on the order of 10 or even sometimes less),

  • “ETKF-N” (Ensemble-Transform Kalman Filter of finite size N), ETKF algorithm of “finite size N”, that doesn’t need inflation that is often required with the other algorithms,

  • “MLEF” (Maximum Likelihood Kalman Filter, see [Zupanski05]), deterministic EnKF algorithm, allowing in addition the consistent treatment of non-linear observation operator,

  • “IEnKF” (Iterative EnKF), deterministic EnKF algorithm, improving treament of operators non-linearities,

  • “E3DVAR” (EnKF 3DVAR, or 3D-Var-Ben), algorithm coupling ensemble and variational assimilation, which uses in parallel a 3DVAR variational assimilation for a single best estimate and an EnKF ensemble algorithm to improve the estimation of a posteriori error covariances,

  • “EnKS” (Ensemble Kalman Smoother), smoothing algorithm with a fixed time lag L.

Without being a universal recommendation, one recommend to use “EnKF” formulation as a reference algorithm, “ETKF-N” ou “IEnKF” formulation for robust performance, and the other algorithms (in this order) as means to obtain a less costly data assimilation with (hopefully) the same quality.

13.7.2. Optional and required commands

The general required commands, available in the editing user graphical or textual interface, are the following:

Background

Vector. The variable indicates the background or initial vector used, previously noted as \mathbf{x}^b. Its value is defined as a “Vector” or “VectorSerie” type object. Its availability in output is conditioned by the boolean “Stored” associated with input.

BackgroundError

Matrix. This indicates the background error covariance matrix, previously noted as \mathbf{B}. Its value is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

EvolutionError

Matrix. The variable indicates the evolution error covariance matrix, usually noted as \mathbf{Q}. It is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

EvolutionModel

Operator. The variable indicates the evolution model operator, usually noted M, which describes an elementary step of evolution. Its value is defined as a “Function” type object or a “Matrix” type one. In the case of “Function” type, different functional forms can be used, as described in the section Requirements for functions describing an operator. If there is some control U included in the evolution model, the operator has to be applied to a pair (X,U).

Observation

List of vectors. The variable indicates the observation vector used for data assimilation or optimization, and usually noted \mathbf{y}^o. Its value is defined as an object of type “Vector” if it is a single observation (temporal or not) or “VectorSeries” if it is a succession of observations. Its availability in output is conditioned by the boolean “Stored” associated in input.

ObservationError

Matrix. The variable indicates the observation error covariance matrix, usually noted as \mathbf{R}. It is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

ObservationOperator

Operator. The variable indicates the observation operator, usually noted as H, which transforms the input parameters \mathbf{x} to results \mathbf{y} to be compared to observations \mathbf{y}^o. Its value is defined as a “Function” type object or a “Matrix” type one. In the case of “Function” type, different functional forms can be used, as described in the section Requirements for functions describing an operator. If there is some control U included in the observation, the operator has to be applied to a pair (X,U).

The general optional commands, available in the editing user graphical or textual interface, are indicated in List of commands and keywords for data assimilation or optimization case. Moreover, the parameters of the command “AlgorithmParameters” allows to choose the specific options, described hereafter, of the algorithm. See Description of options of an algorithm by “AlgorithmParameters” for the good use of this command.

The options are the following:

EstimationOf

Predefined name. This key allows to choose the type of estimation to be performed. It can be either state-estimation, with a value of “State”, or parameter-estimation, with a value of “Parameters”. The default choice is “State”.

Example: {"EstimationOf":"Parameters"}

HybridCostDecrementTolerance

Real value. This key indicates a limit value, leading to stop successfully the optimization process for the variational part in the coupling, when the cost function decreases less than this tolerance at the last step. The default is 1.e-7, and it is recommended to adapt it to the needs on real problems. One can refer to the section describing ways for Convergence control for calculation cases and iterative algorithms for more detailed recommendations.

Example: {"HybridCostDecrementTolerance":1.e-7}

HybridCovarianceEquilibrium

Real value. This key indicates, in hybrid variational optimization, the equilibrium factor between the static a priori covariance and the ensemble covariance. This factor is between 0 and 1, and its default value is 0.5.

Example : {"HybridCovarianceEquilibrium":0.5}

HybridMaximumNumberOfIterations

Integer value. This key indicates the maximum number of internal iterations allowed for hybrid optimization, for the variational part. The default is 15000, which is very similar to no limit on iterations. It is then recommended to adapt this parameter to the needs on real problems. For some optimizers, the effective stopping step can be slightly different of the limit due to algorithm internal control requirements. One can refer to the section describing ways for Convergence control for calculation cases and iterative algorithms for more detailed recommendations.

Example: {"HybridMaximumNumberOfIterations":100}

InflationFactor

Real value. This key specifies the inflation factor in the ensemble methods, to be applied on the covariance or the anomalies depending on the choice of the type of inflation. Its value must be positive if the inflation is additive, or greater than 1 if the inflation is multiplicative. The default value is 1, which leads to an absence of multiplicative inflation. The absence of additive inflation is obtained by entering a value of 0.

Example : {"InflationFactor":1.}

InflationType

Predefined name. This key is used to set the inflation method in ensemble methods, for those that require such a technique. Inflation can be applied in various ways, according to the following options: multiplicative or additive by the specified inflation factor, applied on the background or on the analysis, applied on covariances or on anomalies. The multiplicative inflation on anomalies, that are obtained by subtracting the ensemble mean, is elaborated by multiplying these anomalies by the inflation factor, then by rebuilding the ensemble members by adding the previously evaluated mean. Only one type of inflation is applied at the same time, and the default value is “MultiplicativeOnAnalysisAnomalies”. The possible names are in the following list: [ “MultiplicativeOnAnalysisAnomalies”, “MultiplicativeOnBackgroundAnomalies”, ].

Example : {"InflationType":"MultiplicativeOnAnalysisAnomalies"}

NumberOfMembers

Integer value. This key indicates the number of members used to realize the ensemble method. The default is 100, and it is recommended to adapt it to the needs on real problems.

Example: {"NumberOfMembers":100}

SetSeed

Integer value. This key allow to give an integer in order to fix the seed of the random generator used in the algorithm. By default, the seed is left uninitialized, and so use the default initialization from the computer, which then change at each study. To ensure the reproducibility of results involving random samples, it is strongly advised to initialize the seed. A simple convenient value is for example 123456789. It is recommended to put an integer with more than 6 or 7 digits to properly initialize the random generator.

Example: {"SetSeed":123456789}

SmootherLagL

Integer value. This key indicates the number of smoothing time intervals in the past for the EnKS. This is a number of intervals, not a fixed duration. The default value is 0, which leads to no smoothing.

Example : {"SmootherLagL":0}

StoreSupplementaryCalculations

List of names. This list indicates the names of the supplementary variables, that can be available during or at the end of the algorithm, if they are initially required by the user. Their availability involves, potentially, costly calculations or memory consumptions. The default is then a void list, none of these variables being calculated and stored by default (excepted the unconditional variables). The possible names are in the following list (the detailed description of each named variable is given in the following part of this specific algorithmic documentation, in the sub-section “Information and variables available at the end of the algorithm”): [ “Analysis”, “APosterioriCorrelations”, “APosterioriCovariance”, “APosterioriStandardDeviations”, “APosterioriVariances”, “BMA”, “CostFunctionJ”, “CostFunctionJAtCurrentOptimum”, “CostFunctionJb”, “CostFunctionJbAtCurrentOptimum”, “CostFunctionJo”, “CostFunctionJoAtCurrentOptimum”, “CurrentIterationNumber”, “CurrentOptimum”, “CurrentState”, “ForecastCovariance”, “ForecastState”, “IndexOfOptimum”, “InnovationAtCurrentAnalysis”, “InnovationAtCurrentState”, “SimulatedObservationAtCurrentAnalysis”, “SimulatedObservationAtCurrentOptimum”, “SimulatedObservationAtCurrentState”, ].

Example : {"StoreSupplementaryCalculations":["CurrentState", "Residu"]}

Variant

Predefined name. This key allows to choose one of the possible variants for the main algorithm. The default variant is the original “EnKF” formulation, and the possible choices are “EnKF” (Ensemble Kalman Filter), “ETKF” (Ensemble-Transform Kalman Filter), “ETKF-N” (Ensemble-Transform Kalman Filter), “MLEF” (Maximum Likelihood Kalman Filter), “IEnKF” (Iterative_EnKF), “E3DVAR” (EnKF 3DVAR), “EnKS” (Ensemble Kalman Smoother).

One recommends to try the “ETKF-N” or “IEnKF” variants for a robust performance, and to reduce the number of members to about 10 or less for all variants other than the original “EnKF” formulation.

Example : {"Variant":"EnKF"}

13.7.3. Information and variables available at the end of the algorithm

At the output, after executing the algorithm, there are information and variables originating from the calculation. The description of Variables and information available at the output show the way to obtain them by the method named get, of the variable “ADD” of the post-processing in graphical interface, or of the case in textual interface. The input variables, available to the user at the output in order to facilitate the writing of post-processing procedures, are described in the Inventory of potentially available information at the output.

Permanent outputs (non conditional)

The unconditional outputs of the algorithm are the following:

Analysis

List of vectors. Each element of this variable is an optimal state \mathbf{x}^* in optimization, an interpolate or an analysis \mathbf{x}^a in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

Set of on-demand outputs (conditional or not)

The whole set of algorithm outputs (conditional or not), sorted by alphabetical order, is the following:

Analysis

List of vectors. Each element of this variable is an optimal state \mathbf{x}^* in optimization, an interpolate or an analysis \mathbf{x}^a in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

APosterioriCorrelations

List of matrices. Each element is an a posteriori error correlations matrix of the optimal state, coming from the \mathbf{A} covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: apc = ADD.get("APosterioriCorrelations")[-1]

APosterioriCovariance

List of matrices. Each element is an a posteriori error covariance matrix \mathbf{A} of the optimal state.

Example: apc = ADD.get("APosterioriCovariance")[-1]

APosterioriStandardDeviations

List of matrices. Each element is an a posteriori error standard errors diagonal matrix of the optimal state, coming from the \mathbf{A} covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: aps = ADD.get("APosterioriStandardDeviations")[-1]

APosterioriVariances

List of matrices. Each element is an a posteriori error variance errors diagonal matrix of the optimal state, coming from the \mathbf{A} covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: apv = ADD.get("APosterioriVariances")[-1]

BMA

List of vectors. Each element is a vector of difference between the background and the optimal state.

Example: bma = ADD.get("BMA")[-1]

CostFunctionJ

List of values. Each element is a value of the chosen error function J.

Example: J = ADD.get("CostFunctionJ")[:]

CostFunctionJAtCurrentOptimum

List of values. Each element is a value of the error function J. At each step, the value corresponds to the optimal state found from the beginning.

Example: JACO = ADD.get("CostFunctionJAtCurrentOptimum")[:]

CostFunctionJb

List of values. Each element is a value of the error function J^b, that is of the background difference part. If this part does not exist in the error function, its value is zero.

Example: Jb = ADD.get("CostFunctionJb")[:]

CostFunctionJbAtCurrentOptimum

List of values. Each element is a value of the error function J^b. At each step, the value corresponds to the optimal state found from the beginning. If this part does not exist in the error function, its value is zero.

Example: JbACO = ADD.get("CostFunctionJbAtCurrentOptimum")[:]

CostFunctionJo

List of values. Each element is a value of the error function J^o, that is of the observation difference part.

Example: Jo = ADD.get("CostFunctionJo")[:]

CostFunctionJoAtCurrentOptimum

List of values. Each element is a value of the error function J^o, that is of the observation difference part. At each step, the value corresponds to the optimal state found from the beginning.

Example: JoACO = ADD.get("CostFunctionJoAtCurrentOptimum")[:]

CurrentIterationNumber

List of integers. Each element is the iteration index at the current step during the iterative algorithm procedure. There is one iteration index value per assimilation step corresponding to an observed state.

Example: cin = ADD.get("CurrentIterationNumber")[-1]

CurrentOptimum

List of vectors. Each element is the optimal state obtained at the usual step of the iterative algorithm procedure of the optimization algorithm. It is not necessarily the last state.

Example: xo = ADD.get("CurrentOptimum")[:]

CurrentState

List of vectors. Each element is a usual state vector used during the iterative algorithm procedure.

Example: xs = ADD.get("CurrentState")[:]

ForecastCovariance

Liste of matrices. Each element is a forecast state error covariance matrix predicted by the model during the time iteration of the algorithm used.

Example : pf = ADD.get("ForecastCovariance")[-1]

ForecastState

List of vectors. Each element is a state vector forecasted by the model during the iterative algorithm procedure.

Example: xf = ADD.get("ForecastState")[:]

IndexOfOptimum

List of integers. Each element is the iteration index of the optimum obtained at the current step of the iterative algorithm procedure of the optimization algorithm. It is not necessarily the number of the last iteration.

Example: ioo = ADD.get("IndexOfOptimum")[-1]

InnovationAtCurrentAnalysis

List of vectors. Each element is an innovation vector at current analysis. This quantity is identical to the innovation vector at analysed state in the case of a single-state assimilation.

Example: da = ADD.get("InnovationAtCurrentAnalysis")[-1]

InnovationAtCurrentState

List of vectors. Each element is an innovation vector at current state before analysis.

Example: ds = ADD.get("InnovationAtCurrentState")[-1]

SimulatedObservationAtCurrentAnalysis

List of vectors. Each element is an observed vector simulated by the observation operator from the current analysis, that is, in the observation space. This quantity is identical to the observed vector simulated at current state in the case of a single-state assimilation.

Example: hxs = ADD.get("SimulatedObservationAtCurrentAnalysis")[-1]

SimulatedObservationAtCurrentOptimum

List of vectors. Each element is a vector of observation simulated from the optimal state obtained at the current step the optimization algorithm, that is, in the observation space.

Example: hxo = ADD.get("SimulatedObservationAtCurrentOptimum")[-1]

SimulatedObservationAtCurrentState

List of vectors. Each element is an observed vector simulated by the observation operator from the current state, that is, in the observation space.

Example: hxs = ADD.get("SimulatedObservationAtCurrentState")[-1]