\r\n\texplores a comprehensive overview of the recent theories on flow, transport, industrial applications and processes in porous media. It also extensively discusses sophisticated porous media models which can be used to improve the modeling in practical applications.
\r\n
\r\n\tFeaturing contributions from renowned experts in their respective fields, this book,
\r\n
\r\n\tDescribes the general characteristics of Modelling and Mechanics in Porous Media, such as multiscale modeling of porous media, two-phase flow, and compressible porous media.
\r\n
\r\n\tCovers heat and mass transfer of nanofluids as well as thermal transport in porous media, including natural/forced convection, double-diffusive convection, heat transfer enhancement using nanofluids. \r\n\tAddresses geological applications in porous media, Transport in Porous media in natural resources and applications in the environment. \r\n\tDiscusses relevant work experimental work or numerical techniques wherever applicable. \r\n\tPresents innovative approaches to applying existing porous media technologies to bio-science, biotechnology and biomedical applications. \r\n\t
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Amorim",authors:[{id:"16970",title:"Prof.",name:"Zenilda",middleName:"L.",surname:"Cardeal",fullName:"Zenilda Cardeal",slug:"zenilda-cardeal"},{id:"16984",title:"Dr.",name:"Leiliane",middleName:"C.A.",surname:"Amorim",fullName:"Leiliane Amorim",slug:"leiliane-amorim"},{id:"16985",title:"Prof.",name:"Amauri",middleName:null,surname:"Souza",fullName:"Amauri Souza",slug:"amauri-souza"}]},{id:"13029",title:"Micropollutant Degradation Mechanism",slug:"micropollutant-degradation-mechanism",signatures:"Brigita Tepuš, Irena Petrinić and Marjana Simonič",authors:[{id:"15110",title:"Dr.",name:"Irena",middleName:null,surname:"Petrinic",fullName:"Irena Petrinic",slug:"irena-petrinic"},{id:"15704",title:"Dr.",name:"Brigita",middleName:null,surname:"Tepuš",fullName:"Brigita Tepuš",slug:"brigita-tepus"},{id:"16583",title:"Dr.",name:"Marjana",middleName:null,surname:"Simonič",fullName:"Marjana Simonič",slug:"marjana-simonic"}]},{id:"13030",title:"Bacterial-Degradation of Pesticides Residue in Vegetables during Fermentation",slug:"bacterial-degradation-of-pesticides-residue-in-vegetables-during-fermentation",signatures:"Aslan Azizi",authors:[{id:"15701",title:"Dr.",name:"Aslan",middleName:null,surname:"Azizi",fullName:"Aslan Azizi",slug:"aslan-azizi"}]},{id:"13107",title:"Interpretation and Modelling of Environmental Behaviour of Diverse Pesticides by Revealing Photodecomposition Mechanisms",slug:"interpretation-and-modelling-of-environmental-behaviour-of-diverse-pesticides-by-revealing-photodeco",signatures:"Attila Kiss and Diána Virág",authors:[{id:"14950",title:"Dr.",name:"Attila",middleName:null,surname:"Kiss",fullName:"Attila Kiss",slug:"attila-kiss"},{id:"14973",title:"Dr.",name:"Diána",middleName:null,surname:"Virág",fullName:"Diána Virág",slug:"diana-virag"}]},{id:"13031",title:"Degradation of Organochlorine and Organophosphorus Pesticides by Photocatalysis: Chlorpiryfos and Endosulfan Case Study",slug:"degradation-of-organochlorine-and-organophosphorus-pesticides-by-photocatalysis-chlorpiryfos-and-end",signatures:"Rosalina González Forero",authors:[{id:"14241",title:"Dr.",name:"Rosalina",middleName:null,surname:"Gonzalez Forero",fullName:"Rosalina Gonzalez Forero",slug:"rosalina-gonzalez-forero"}]},{id:"13032",title:"Advanced Oxidation Processes (AOPs) for Removal of Pesticides from Aqueous Media",slug:"advanced-oxidation-processes-aops-for-removal-of-pesticides-from-aqueous-media",signatures:"Marco A. Quiroz, Erick R. Bandala and Carlos A. Martínez-Huitle",authors:[{id:"15324",title:"Dr.",name:"Marco Antonio",middleName:null,surname:"Quiroz Alfaro",fullName:"Marco Antonio Quiroz Alfaro",slug:"marco-antonio-quiroz-alfaro"},{id:"16897",title:"Dr.",name:"Erick Roberto",middleName:null,surname:"Bandala González",fullName:"Erick Roberto Bandala González",slug:"erick-roberto-bandala-gonzalez"},{id:"16898",title:"Dr.",name:"Carlos Alberto",middleName:null,surname:"Martínez Huitle",fullName:"Carlos Alberto Martínez Huitle",slug:"carlos-alberto-martinez-huitle"}]},{id:"13033",title:"Low-Cost Sorbent for Removing Pesticides during Water Treatment",slug:"low-cost-sorbent-for-removing-pesticides-during-water-treatment",signatures:"Katarzyna Ignatowicz",authors:[{id:"15053",title:"Prof.",name:"Katarzyna",middleName:null,surname:"Ignatowicz",fullName:"Katarzyna Ignatowicz",slug:"katarzyna-ignatowicz"}]},{id:"13034",title:"Influence of the Activated Carbon Nature and the Aqueous Matrix on the Pesticides Adsorption",slug:"influence-of-the-activated-carbon-nature-and-the-aqueous-matrix-on-the-pesticides-adsorption",signatures:"Natividad Miguel, María P. Ormad, Rosa Mosteo, Jorge Rodríguez and José L. Ovelleiro",authors:[{id:"14648",title:"Dr.",name:"Natividad",middleName:null,surname:"Miguel",fullName:"Natividad Miguel",slug:"natividad-miguel"},{id:"14649",title:"Prof.",name:"María P.",middleName:null,surname:"Ormad",fullName:"María P. Ormad",slug:"maria-p.-ormad"},{id:"14650",title:"Dr.",name:"Rosa",middleName:null,surname:"Mosteo",fullName:"Rosa Mosteo",slug:"rosa-mosteo"},{id:"14651",title:"Prof.",name:"Jorge",middleName:null,surname:"Rodríguez",fullName:"Jorge Rodríguez",slug:"jorge-rodriguez"},{id:"14652",title:"Dr.",name:"José L.",middleName:null,surname:"Ovelleiro",fullName:"José L. Ovelleiro",slug:"jose-l.-ovelleiro"}]},{id:"13035",title:"Adsorption Properties of Sediments for Pesticides: Investigation with Supercritical Fluid Extraction and Gas Chromatograph Mass Spectrometry",slug:"adsorption-properties-of-sediments-for-pesticides-investigation-with-supercritical-fluid-extraction-",signatures:"Hiroaki Chikushi, Natsuko Yoshida and Kei Toda",authors:[{id:"14786",title:"Dr.",name:"Kei",middleName:null,surname:"Toda",fullName:"Kei Toda",slug:"kei-toda"},{id:"16534",title:"Dr.",name:"Hiroaki",middleName:null,surname:"Chikushi",fullName:"Hiroaki Chikushi",slug:"hiroaki-chikushi"}]},{id:"13036",title:"Sorption of Pesticides on Natural Geosorbents",slug:"sorption-of-pesticides-on-natural-geosorbents",signatures:"Jean-Pierre Gagné, Bruno Gouteux, Youssouf Djibril Soubaneh, and Jean-Rock Brindle",authors:[{id:"14741",title:"Dr.",name:"Jean-Pierre",middleName:null,surname:"Gagné",fullName:"Jean-Pierre Gagné",slug:"jean-pierre-gagne"},{id:"14761",title:"Dr.",name:"Jean-Rock",middleName:null,surname:"Brindle",fullName:"Jean-Rock Brindle",slug:"jean-rock-brindle"},{id:"16683",title:"Dr.",name:"Youssouf Djibril",middleName:null,surname:"Soubaneh",fullName:"Youssouf Djibril Soubaneh",slug:"youssouf-djibril-soubaneh"},{id:"16684",title:"Dr.",name:"Bruno",middleName:null,surname:"Gouteux",fullName:"Bruno Gouteux",slug:"bruno-gouteux"}]},{id:"13037",title:"Pesticides as a Waste Problem with Examples from Norway",slug:"pesticides-as-a-waste-problem-with-examples-from-norway",signatures:"Ketil Haarstad",authors:[{id:"16783",title:"Dr.",name:"Ketil",middleName:null,surname:"Haarstad",fullName:"Ketil Haarstad",slug:"ketil-haarstad"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"72778",title:"Adaptive Filter as Efficient Tool for Data Assimilation under Uncertainties",doi:"10.5772/intechopen.92194",slug:"adaptive-filter-as-efficient-tool-for-data-assimilation-under-uncertainties",body:'
1. Introduction
In this chapter, the adaptive filter (AF) is considered as a computational device that yields estimates of the system state by minimizing recursively (in time) the error between the predicted output of the device and its observed signal in real time. As the main objective of the AF is to produce estimates of the state in high-dimensional systems (HdSs), we shall focus the attention on the mathematical form of the AF in a state-space form as that used in the Kalman filter (KF) [1]. In this chapter, the HdS is referred to as a system whose state dimension is of order O107−O108.
The assimilation problem in this chapter is formulated as a standard filtering problem. For simplicity, let the dynamical system be described by the equation
xt+1=Φxt+wt,x0=x0,t=0,1,…E1
where xt is the system state at the t time instant. At each time instant t, we are given the observation for the system output
zt+1=Hxt+1+vt+1,t=0,1,…E2
In (1) and (2), wt is the model error (ME), vtis the observation error (ObE), and Φ represents the system dynamics. In general, the system (1) and (2) may be nonlinear with Φx=fx, Hx=hx. The filtering problem for a partial observed dynamical system (1) and (2) is to obtain the best possible estimate for the state xtat each instant t, given the set of observations Z1:t=z1…zt.
There exist different techniques to solve estimation problems. The simplest approach is related to linear estimator [2], since it requires only first two moments. Linear estimation is frequently used in practice when there is a limitation in computational complexity. Among others, the widely used methods are maximum likelihood, least squares, method of moments, the Bayesian estimation, minimum mean square error (MMSE), etc. For more details, see [3].
There are limitations of optimal filters. In practice, the difficulties are numerous: the statistics of signals which may not be available or cannot be accurately estimated; there may not be available time for statistical estimation (real-time); the signals and systems may be non-stationary; memory required and computational load may be prohibitive. All these difficulties become insurmountable, especially for HdSs.
In order to deal with real-time applications, the AFs appear to be a valuable tool in solving estimation problems when there is no time for statistical estimation and when we are dealing with non-stationary signals and/or systems environment. They can operate satisfactorily in unknown and possibly time-varying environments without user intervention. They improve their performance during operation by learning statistics from current signal observations. Finally, they can track variations in the signal operating environment [4].
It is well-known that the MMSE estimator in the class of Borel measurable (with respect to (wrt) Z1:t) functions is given by the conditional mean
x̂t=Ext/Z1:tE3
Under standard conditions, related to the noise sequences wt,vt (Gaussian i.i.d.—identically independent (temporal) distributed), the estimate (3)x̂t for xt can be obtained from the KF in the recursive form
x̂t+1=x̂t+1/t+Kζt+1,E4
x̂t+1/t=Φx̂t,ζt+1=zt+1−x̂t+1/tE5
Kt=MtHTHMtHT+R−1E6
Mt+1=ΦPtΦT+QE7
Pt=I−KtHMtI−KtHT+KtRKTtE8
In (4)–(8), Q,R are the covariance matrix for wt and vt, respectively. One sees that K=KM—the gain matrix, is a function of M≔Mt+1—the error covariance matrix (ECM) for the state prediction error (PE) et+1/t which is defined as et+1/t≔xt+1−x̂t+1, and ζt+1 is known as innovation vector. Note from (7) and (8) that the ECM Mt+1 can be found by solving the matrix nonlinear Algebraic Riccati equation (ARE). Generally speaking, a solution of the ARE is not unique. Conditions must be introduced for ensuring an existence of a unique non-negative definite solution [5]. It is remarkable that the ECMs P,M in (7) and (8) do not depend on observations; therefore, they can be computed in advance, offline, given the system matrices and the noise covariances. The same remark is valid for the gain matrix K in (6). In contrast, the gain in the AF is observation-dependent [6] (see Section 2).
Under the most favorable conditions (perfect knowledge of all system parameters and noise statistics), for a dynamical system with dimension of order 107–108, it is impossible to solve the ARE (due to computational burden), not to say about storing M,P. To overcome these difficulties, the AF is proposed. Mention that the KF is also an MMSE filter in the complete Hilbert space of random variables, Eξ2<∞, with scalar product ξ1ξ2H=E<ξ1ξ2> and the norm ξ=Eξ21/2.
For the nonlinear models, there are KF variants, among those are the extended KF (EKF) [7], the unscented KF (UKF, [8]), and the Ensemble Kalman filter (EnKF, [9]). In the EnKF, the ECM is a sampled ECM whose samples are generated using samples of the state variable, and consequently the ECM in the KF becomes a sampled ECM. For an example of application of the EnKF for data assimilation in geophysical data assimilation with high dimensional model, see [10]. Another class of ensemble filtering technique is a class of particle filters (PF, [11]). The basic idea of the PF (also the EnKF) is to use a discrete set of weighted n particles to represent the distribution of xt, where the distribution is updated at each time by changing the particle weights according to their likelihoods.
Despite a possible implementation of the KF variants, they might still be seriously biased because the accuracy of the KF update requires linearity of the observation function and Gaussianity of the distribution of system state xt. In reality, the KF (4)–(8) may be biased and unstable, even divergent [12]. Today, the PF algorithms are ineffective for HdS data assimilation.
In this chapter, we shall show how the AF can be efficient in dealing with uncertainties existing in the filtering problem (1) and (2). In Section 2, a brief outline of the AF is given. The main features of the AF, which are different to those of the KF, are presented. This concerns the optimal criteria, stabilizing gain structure, optimization algorithms. Section 3 shows in detail how the AF is capable of dealing with uncertainties in the specification of ME covariance. The hypothesis on a subspace of ME is presented in Section 4 from which one sees how one can make order reduction for representing the bias and ME covariance. Simple numerical examples on one- and two-dimensional systems are given in Section 5 to illustrate in details the differences between the AF and the traditional KF. Numerical experiments on low and high dimensional systems are given in Section 6 to demonstrate how the AF algorithm works. The performance comparison between the AF and KF, for both situations of perfect knowledge of ME statistics and that with ME uncertainties, is also presented. Conclusions and perspectives of the AF are summarized in Section 7.
2. Adaptive filter
The AF is originated from [13]. It is constructed for estimating the state of a dynamical system based on partially observed quantities related in some way to the system state. As reported before, for linear systems contaminated by Gaussian noise, the MMSE estimate can be obtained by the KF. Since publication of [1] in 1960, an uncountable number of works are done for solving engineering problems by KF, in all engineering fields, as well as many modifications have been proposed. The reasons for the need in modification of the KF are numerous, but mostly related to nonlinear dynamics, parameter uncertainties in specification of system parameters, bias of ME, unknown statistics of ME, model reduction. With the rapid progress of computer technology (computer power, memory, …), various simplified versions of KF are suggested for solving filtering (or data assimilation) for HdSs, in particular, in meteorology and oceanography.
Direct application of the KF to HdSs is impossible due to the limit in computer power, memory, and computational time. In particular, the KF requires to solve the matrix AREs (7) and (8) for computing ECMs Mt,Pt. Storing such matrices is impossible, not to say on computational time.
Different simplified approaches are proposed for overcoming difficulties in the application of the KF. The example of successful tool for solving data assimilation problems in HdSs is the EnKF [9]. In the EnKF, an ensemble of error samples, of small size, is generated on the basis of model states to approximate the ECMs. In practice of data assimilation for HdSs, it is possible to generate only ensembles of moderate sizes (of order O100) by model integrations over the assimilation window (time interval between two successive arrivals of observations) since one such integration takes several hours! The other approach like PF is based on sampling from conditional distributions. Theoretically, this approach is more appropriate for nonlinear problems because no linearization is required as in the EKF (Extended KF based on linearization technique). However, even for filtering problems with state dimensions of order O10, relatively large ensembles of size O10000 would be required in order to produce reasonably good performance.
The AF in [13] is based on the different idea. Here, no linearization is required for nonlinear filtering problems. For the problem (1) and (2), the filter is of the form (4) and (5) but the gain K=Kθis assumed to be of a given stabilizing structure [6]. It means that K is parametrized by some vector of unknown parameters θ∈Θ so that the filter (4) and (5) with the gain Kθ,∀θ∈Θis stable. It is well-known that under mild conditions, the solution of the ARE will tend (quickly) to stationary solution M∞ and so the gain (6), to the stationary gain K∞. Moreover, the innovation ζt+1=zt+1−ẑt+1/t, representing the error for the output predictionẑt+1/t≔Hx̂t+1/t=HΦx̂t, is unbiased and of minimum variance. This fact leads to the idea to seek the optimal vector θby minimizing
Jθ≔EΨζ→argminθE9
here E. denotes the average in a probabilistic sense. For stationary systems (1) and (2), if we assume the validity of the ergodic hypothesis, the average value in a probabilistic sense, expressed in (7), is almost everywhere equivalent to the time average (for large time of running the dynamical system). The optimal θ∗ can be found by solving the equation
∇θJθ=∇θEΨζ=E∇θΨζ=0E10
A stochastic approximation (SA) algorithm for solving (10) can be written out
θt+1=θt−γt∇θtΨζt+1E11
Conditions related to the sequence of positive scalar γt for ensuring a convergence θtin the procedure (11) are
γt>0,∑t=0∞γt=∞,∑t=0∞γ2t<∞E12
One of the most advantages of the SA algorithm (11) is that, instead of computing the gradient of the cost function (9) (which requires knowledge of probability distribution), the algorithm (11) is based on the knowledge of only the gradient of sample cost function Ψ (wrt to θ) which can be easily evaluated numerically.
Comment 2.1. Generally speaking, the convergence rate of the algorithm (11) and (12) is O1/t. It is possible to improve the convergence rate of the SA by averaging of the iterates,
Comment 2.2. For high HdSs, even with θ being of moderate dimension, instead of the algorithm (12) or (13), the SPSA (Simultaneous Perturbation Stochastic Perturbation) algorithms in [15, 16] are of preference. That is due to the fact that integration of HdS over the assimilation window is very expensive. These algorithms generate stochastic perturbation δθ=δθ1…δθnT with components as Bernoulli i.i.d. realizations. Each ithcomponent of the gradient-like (pseudo-gradient) vector is computed as the divided difference δΨ/δθi where Ψ≔Ψθ+δθ−Ψθ. This allows to evaluate the gradient-like vector by only two or three time integration of the numerical model.
For details on the SPSA algorithm and its convergence rate, see [15, 16].
3. Covariance uncertainties and AF
3.1 Covariance uncertainties
3.1.1 Adjoint approach
As seen from (11) and (12), implementation of SA algorithms is much simpler for searching optimal gain parameters compared to the other optimization methods. The SA algorithms require only numerical computing derivatives ∇θΨζt+1 of the sample cost function Ψζt+1 wrt θ evaluated at θt and γtis a scalar which can be chosen a priori, for example, as γt=1t. That is possible due to introducing the ergodic hypothesis on of the system (1) and (2) from which there exists an asymptotic optimal gain
First, consider the situation when the vector of parameters consists of are all elements of K,θ=K. Compute the innovation vector,
ζt+1=zt+1−ẑt+1/t=zt+1−HΦx̂t
x̂t=x̂t−1/t+Kθζt
δθζt+1=−HΦδθx̂t=−HΦδθKθζt=−HΦδKζt.
δKΨζt+1=δK<ζt+1,ζt+1>=2<ζt+1,δKζt+1>
=−2<ζt+1,HΦδKζt>=−2<ΦTHTζt+1,δKζt>.
Let us compute derivatives of the sample cost function Ψ wrt the elements Kij of the gain K. To do so, one needs to integrate the adjoint operator ΦT s.t. the forcing f≔−HTζt+1 which yields ψ≔−ΦTHTζt+1 and hence
δΨ/δKij=−ψiζtj,E14
here ψi is theithcomponent of ψ,ζtj the jthcomponent of ζt. The AF now takes the form
x̂t+1=x̂t+1/t+Kζt+1,E15
x̂t+1/t=Φx̂t,ζt+1=zt+1−Hx̂t+1/t,E16
Kt+1=Kt−γt∇KΨζt+1,E17
where ∇KΨζt+1 is the gradient vector whose components are computed by (14). In the AF (15)–(17), no matrix ARE (see (7) and (8) in the KF) is involved. The AF (15)–(17) is quite realizable for HdSs, since at each assimilation instant we need to integrate only the direct model to produce the forecast (16) and (eventually) an adjoint model over the assimilation window for computing ∇KΨζt+1 [13].
Remark that in the form (14) the adjoint operator ΦT would be available to implement the AF. It is well-known that construction of numerical code for ΦT is a very difficult and heavy task, especially for meteorological and oceanic numerical models which are HdSs and nonlinear (linearization is required).
A comparison study of the AF with other assimilation methods is done in [17]. Compared to the AF, the widely used variational method (VM) minimizes the distance between the observations available (for example, the observations of the whole set Z1:T) and the outputs of the dynamical system. This optimization problem is carried out in the phase space, hence is very difficult and expensive. Theoretically, a simplification is possible subject to (s.t.) the condition of linearity of the dynamical system: in this case, one can reformulate the VM minimization problem as searching the best estimate for the initial system state x0. For HdSs, to ensure a merely high quality estimate for x0, it is necessary: (i) to take the observation window as large as possible; (ii) to parameterize the initial state by some parameters (using a slow manifold, for example). Iterative minimization procedures require usually O10 iterates involving integrating the direct and adjoint models over the window 1T. For an unstable dynamics, integration of direct and adjoint equations over a long period naturally amplify the initial errors during assimilation process. For a more detailed comparison between the AF and VM, see [17].
Thus, if the ergodic conditions hold, there exists an optimal stationary gain and the AF in limit will approach to the optimal one in the given class of stable filters. It is important to emphasize that up to this point, no covariance matrices Q,R are specified. It means that the AF in the form (12) is robust to uncertainties in the specification of the covariances of the ME and ObE.
3.2 Stability of the AF
One of the main features of the AF is related to its stability.
For simplicity of presentation, in the previous section, the AF algorithm is written out under the assumption (13). In practice, application of the AF in the form (13) is not recommended since instability may occur. It is easy to see that the transition matrix of the filter is given by
L=I−KHΦE18
It is evident that if we do not take care on the structure of K, varying stochastically all elements of K can lead to instability of L and the filter will be exploded. Moreover, for HdSs, the number of elements of K is very large. It is therefore primordial to choose a parametrized stabilizing structure for K (depending on θ) to ensure a stability of L and reducing a number of tuning parameters. This question is addressed in [6]. One of possible structures for K is of the form
Kt=PrΘKe,Ke≔MeHeTHeMeHeT+αI−1,He≔HPrE19
where Pr∈Rnxr is a matrix with dimensions n×r, r is the dimension of the reduced space (equal to the number of unstable eigenvectors (EiVecs)of Φ), the matrix Me is a strictly positive symmetric definitive playing the role of the ECM in the reduced space Re, Θ is a diagonal matrix with diagonal elements θi whose values belong to ϵ2−ϵ, i.e. θi∈ϵ2−ϵ, with ϵ∈01 whose value depends on the modulus of the first stable eigenvalue (EiV) [6]. We will refer to the filter s.t. (19) with Θ=Id (Id is the identity matrix of appropriate dimension) as a nonadaptive filter (NAF). In the AF, the parameters θi are adjusted each time when a new observation arrives, to minimize the cost function (9). Thus θi is a time-varying function. As to the matrix Pr, its choice is important to ensure a filter stability. One simple and efficient procedure (called Prediction Error Sampling Procedure—PeSP) to generate Pr is to use the power orthogonal iteration method [18] which allows to compute real leading Schur vectors (SchVecs) of Φi. The advantage of using the SchVecs compared to the EiVecs, is that they are real and their computation is stable. It is seen that the optimal AF is found in a class of stable filters which is stable even for an unstable numerical model. As to the VM, the optimal trajectory is found on the basis of only the numerical model with the initial state as a control vector. It means that for unstable dynamics, the errors in the forcing or numerical errors arising during computations will be amplified and lead to large estimation error growth. More seriously, the VM requires a large set of observations and large number of iterations (i.e., many forward and backward integrations of the direct and adjoint models) which naturally leads to increase of estimation error too.
3.3 On improving the initial gain
Consider the gain structure (19). Suppose that Me has been chosen in agreement with the required stability conditions. Before tuning the parameters θi to minimize the cost function (9), remark that stability of the filter is still ensured for the following gain:
Kt=PrΛΘKe,E20
where Λ=diagλ1…λr,λk∈01 since then for Γ=ΛΘ=diagγ1…γr it implies γt=λtθt∈ϵ2−ϵ, where ϵ∈01. Writing the equation for the filtered error (FE) eft≔x̂t−xtone sees that the matrix L in (18) also represents the transition of the FE eft. It means that it is possible to choose a more optimal initial gain by solving, for example, the minimization problem
JoΛ=LΛ22→minΛE21
The problem (21) is solved without using the observations, hence it is offline. Once the optimal Λ∗ has been found, the standard AF is implemented s.t. the filter gain
Kt=PrΘKe∗,Ke∗≔Λ∗KeE22
It is seen that using the structure (22) this optimization procedure does not require the information on the ME statistics.
4. Joint estimation of state and model error in AF
The previous section shows how the AF is designed to deal with the difficulty in specification of covariances of the ME and ObE. This is done without exploiting a possibility to determine, more or less correctly, a subspace for the ME. If such a subspace can be determined without major difficulties, it would be beneficial for better estimating the AF gain and improving the filter performance. In [19], the hypothesis of the structure of the ME has been introduced and a number of experiments have been successfully conducted.
There is a long history of joint estimation of state and ME for filtering algorithms, in particular, with the bias and covariance estimation. One of the most original approaches, dealing with the treatment of bias in recursive filtering (known as bias-separated estimation—BSE), is carried out by Friedland in [20]. He has shown that the MMSE state estimator for a linear dynamical system augmented with bias states can be decomposed into three parts: (1) bias-free state estimator; (2) bias estimator; and (3) blender. This BSE approach has the advantage that it requires fewer numerical operations than the traditional augmented-state implementation and avoids numerical ill-conditioning compared to the case of bias-separated estimation by filtering technique.
It is common to treat the bias as part of the system state and then estimate the bias as well as the system state. There are two types of ME—deterministic (DME) and stochastic (SME). Generally speaking, a suitable equation can be introduced for the ME. In the presence of bias, under the assumption on constant b, instead of (1) one has
xt+1=Φxt+bt+wt,bt+1=bt,t=0,1,2…E23
To introduce a subspace for the variables wt,bt the SME and DME in (23), let
w=Gwρ,b=Gbd
Gw∈Rn×nw,Gb∈Rn×nd,n≥nw,n≥nbE24
Generally speaking, Gw,Gb are unknown, and finding reasonable hypothesizes for them is desirable but not self-evident. In [19], one hypothesis for Gw,Gb has been introduced (it will be referred to as Hypothesis on model error—HME).
The information on Gw,Gb, given in (25), allows to better estimate the DME b and SME w for improving the filter performance, especially for nb < n, nw < n in a HdS setting. The difficulty, encountered in practice of operational forecasting systems, is that (practically) nothing is given a priori on the space of the ME values. To overcome this difficulty, one simple hypothesis has been introduced in [19]. This hypothesis is postulated by taking into consideration the fact that for a large number of data assimilation problems in HdSs, the model time step δt (chosen for ensuring a stability of numerical scheme and for guaranteeing a high precision of the discrete solution) is much smaller than Δt—the assimilation window (time interval between two successive observation arrivals).
Suppose that Δt=naδt where na is a positive integer number.
Hypothesis (on the subspace of ME—HME) [19]. Under the condition that na is relatively large, the ME belongs to the subspace spanned by all unstable and neutral EiVecs (or SchVecs) of the system dynamics Φ.
5. Simple numerical examples
5.1 One-dimensional system
To see the difference between the AF and the KF in doing with ME uncertainties, introduce the one-dimensional system
xt+1=Φxt+wt,zt+1=hxt+1+vt+1,t=0,1,…E25
In (25), Φ is the unique eigenvalue (also the singular value) of the system dynamics.
For simplicity, let Φ=1,h=1. This corresponds to the situation when the system is neutrally stable. The filter fundamental matrix (18) now is LK=1−K which is stable if K∈02. For the KF gain (4)–(8), as Kkft=MkftMkft+R we have Kkft∈01, Mkft is the solution of (7). That is true for any Mkft≥0, R>0. It means then the KF is stable. Mention that if Q>0 always Mkft>0. In general, Kkft=MkftMkft+R+ where A+ is the pseudo-inverse of A [21].
For the AF, we have in this case Pr=1. Consider the gain Kafθ≔PrθKe, where Ke is the gain of the form Ke=MeMe+R, Me>0, R>0, Me is constant. We have then for the NAF (θ=1)0<Ke < 1 and Knaf=Ke.
For the AF, the transition matrix (18) reads Lafθ=1−θKe. For θ∈02, ∣Lkfθ∣∈01, Kafθ∈02 and the AF is stable. It is evident that there is a larger margin for varying the gain in the AF than that in the KF since Kkft∈01. One sees that the stationary KF is a member of the class of stable AFs (19). The performance of AF is optimized by solving the problem (9) using the procedure (11) and (12) or SPSA algorithms (Comment 2.2).
Let Φ<1, i.e., the system (1) is stable. The results in (i) are valid for the AF structure. In this situation, the filter is stable even for Kaf=0.
Let Φ>1—the system (1) is unstable. Consider two situations (a) Φ>1; (b) Φ<−1. As before Pr=1.
For Φ>1we have
Φ−1KeΦ<θ<Φ+1KeΦE26
In particular, when Φ→1, approximately θ∈02Ke. When Q≫R (that is usually in practice), approximately θ∈02 as in the situation (i). For large Φ≫1, Φ−1KeΦ→1Ke (left-hand limit), Φ+1KeΦ→1Ke (right-hand limit) and there remains no margin for varying θ (or Q≫R) and Kaf→1. It is important to emphasize that as Ke is chosen by designer, we can define the interval for varying θ if the amplitude of Φ is more or less known. In practice, one can vary θ∈ϵ2+ϵ with small ϵ>0 for Φ close to 1, and with ϵclose to 1 for large Φ.
For Φ<−1we have
Φ+1KoΦ<θ<Φ−1KoΦE27
It is seen from (27) that when Φ→−1, approximately θ∈02Ke. As for the situation Φ≪−1, Φ+1KoΦ→1Ko (left-hand limit), Φ−1KoΦ→1Ko (right-hand limit) when Q≫R approximately θ→1Kohence Kaf→1.
It is important to stress that the KF gain is computed on the basis of Q and R (under the condition that the statistics of the initial state will be forgotten as t becomes large); whereas, the gain of the AF is updated on the basis of samples of the innovation vector. It means that the KF is optimal in the MMSE sense (under the condition of exact knowledge of the required statistics) whereas the AF is optimized during the assimilation process using PE realizations of the system output (innovation vector). The KF gain can be computed in an offline fashion, whereas the AF gain is a function of observation and computed in online.
5.2 Two-dimensional system: specification of covariances
5.2.1 Stable filter
To see the role of the correction subspace RPr in ensuring a stability of the AF, let us consider the system (1) and (2) s.t.
Φ=diagΦ11Φ22,H=diag11E28
Consider the AF with the gain (19) s.t. Pr=Id, i.e., two columns of Pr are in fact the EiVecs associated with two EiVs Φ11 and Φ22. Let us denote the AF gain Kaf=PrΘMHTHMHT+R−1=ΘKe with Ke≔MeHTHMeHT+R−1 which is structurally identical to that of the KF. For the nonadaptive filter Knaf=Ke and with the choice Me=diagM11M22, taking into account (28) one gets
Knaf=diagK11K22,Kii=MiiMii+Ri,i=1,2E29
Lnaf=diagl11l22,lii=1−MiiMii+RiΦ/ii,i=1,2E30
The filter transition matrix (30) is obtained on the basis of Lnaf=I−KnafHΦ and the assumption (29). It is easily to see that Lnaf has two EiVs, λi=lii,i=1,2 where lij est. the ij element of Lnaf.
Stability of the filter depends on the condition ∣lii∣<1, i=1,2. For Mii>0,Ri>0, if Φii is stable or neutrally stable, i.e. ∣Φii∣≤1,i=1,2, we have ∣lii∣<1. For unstableΦii (∣Φii∣>1,i=1,2), the filter is unstable if Φii>Mii+RiRi (situation Φii>1) or Φii<−Mii+RiRi (situation Φii<−1). These conditions should be taken into account when the EiVs of Φ are large.
From (31) conditions for ∣lii∣<1 can be obtained as done in Section 5.1 with the one-dimensional system since lii,i=1,2 are independent one from another. The length of the interval Iifor varying θi depends on the value of Φii (see (26)).
This example shows that for Pr=Id, it is always possible to construct a stable AF whatever are the EiVs of Φ (stable or unstable). There are some constraints for Mii (they are positive) and for Ri (small positive). Optimality of the AF is obtained by searching recursively (in time) the optimal θi during assimilation process. Thus, in the AF, a correct specification of ME and ObE statistics (second order) is not important as happens in the KF.
5.2.2 Unstable filter
Consider the situation when Pr is constructed from only one vector. Let Pr=10T—the EiVec associated with Φ11 (the results remain the same if we choose Pr=01T—the EiVec associated with Φ22).
Φ=diagΦ11Φ22,∣Φ11∣>1,Φ22>1,H=diag11E32
We show now that the filter with the gain (19) is unstable. We have (for Θ=Id),
if we put R=αI. For Lnaf=I−KHΦ, taking into account (32), (33) it implies
Lnaf=αΦ11me+α00Φ22E34
As αme+α can be made as small as desired by choosing small α>0, the first EiV l11=αΦ11me+α can be made stable. However, the second EiV in (34)l22=Φ22>1 is unstable. It implies that the filter with the gain (19) s.t. Pr=10T is unstable. This happens even for Θ≠Id. It means that when the projection subspace RPr does not contain all unstable and neutral EiVecs of the system dynamics, it is impossible to guarantee a stability of the filter.
5.3 Two-dimensional system: estimation of ME
Consider the filtering problem (1) and (2), the dynamical system (1) describes a sequence of system states at time instants t=0,1,… when the observations are available. It means that Φ represents the transition of system state over the (observation) time window Δt separating arrivals of two successive observations. In practice, the interval Δt is much larger than the model time stepδt which is the step size in approximating the temporal derivative. The choice of δt is important for guaranteeing a stability of discretized scheme and having high is important for guaranteeing a stability of discretized scheme and having high precision of the discretized solution (wrt the continuous solution). We have then ΔT=naδt, where na is a relatively large positive integer. For example, in the HYCOM model at SHOM (French marine) for the Bay of Biscay configuration, the interval Δt between two observation arrivals is 7 days which is equivalent to integrating 1200 model time steps δt. It means na=1200. Symbolically we have then the equations for model time step integration
x′τ+1=Φ′x′τ+ψ′τ,ψ′τ≔b′τ+w′τE35
In (35), Φ′ represents the integration of numerical model over one model time step δt. Hence
Φ=Φ′naE36
The contribution of ψ′τ, over the assimilation window t−1t (for simplicity and without loss of generality, one supposes t−1≔0,t≔n_a) is
ψt=bt+wt,bt≔∑τ=0naν1τ1,ν1τ≔Φ′na−1−τb′τ,
wt≔∑τ=0naν2τ,ν2τ≔Φna−1−τw′τ,Φ′−1≔0E37
The HME in Section 4 says that the SME wt and DME bt, as functions of na, belong to the subspace spanned by leading EiVs (or SchVecs) of Φ for a relatively large na. The initial filtering problem now has the form (1) and (2) s.t. (36) and (37) where t=τ/na.
To illustrate this HME, continue the two-dimensional system in Section 5.2.2 and suppose that ∣Φ′11>1Φ′22<1. Applying HME in this case is equivalent to saying that the values of MEs bt,wt, approximately, belong to the subspace Ru1spanned by the first EiVec u1=col10, associated with the EiV Φ′11. Here y=coly1…yn denotes the vector-column with components y1,…,yn. It follows that the covariance matrix of wt is assumed to be of the form Q=σw2u1u1T and bt—of the structure bt=cu1, c is a scalar to be estimated. For the algorithm of joint estimation of state and bias (in term of c), see [19].
6. Simulation results
6.1 One-dimensional system
In this section, the filtering problem (25) in Section 5.1 is considered s.t.
Φ=0.99,H=1.02,Q=0.09,R=0.01.
The true system states and observations are simulated using the initial state x0=1 and wt,vt are zero mean Gaussian mutually uncorrelated and temporal uncorrelated sequences.
To see the performance of the AF, unknown system states are estimated on the basis of the AF algorithm. To obtain a reference, the standard KF is also implemented for solving this filtering problem. In the filtering algorithms, the estimate of the initial state is x̂0=2. The gain Knaf in the NAF is taken as that of the KF at t=0, i.e., Knaf=Kkf0.
Figure 1 shows the temporal evolution of the parameters θmt during assimilation process.
Figure 1.
Temporal evolution of the parameter θmt in the AF gain.
The gains in the KF and AF during the assimilation process are displayed in Figure 2. Mention that the KF gain is computed s.t. true statistics Q,R. In the AF, θmt has been used for computation of the AF gain, i.e., Kaf=θmtK. From Figure 2, one sees that initialized by the same value, the two gains become different during assimilation process. The KF gain has reached a stationary regime very quickly.
Figure 2.
Temporal evolution of gains in KF and AF during data assimilation.
The mean temporal RMS (root mean square) of the innovation is shown in Figure 3. It is interesting to remark that no significant difference is observed between two curves and a slightly better performance is produced by the KF.
Figure 3.
RMS of innovation produced by the KF and AF.
In Figure 4, we show RMS of the state FE produced by the KF and AF under the condition that the variance Q is known exactly. One sees that the KF, as expected, produces the best results.
Figure 4.
RMS of the state FE produced by the KF and AF under the condition that the variance of ME is known exactly. It is seen that when the ME is correctly specified, the KF behaves better than the AF.
Figure 5 shows the RMS of FE as a function of the variance Q. Here, the value of Q varies from 0.1 to 1.9. Note that the true value of Q is 0.1. The red curve represents the RMS of FE produced by the KF at the end of the assimilation period (as a function of Q). The green curve has the same meaning, but for the FE produced by the AF. It is interesting to note that when Q is correctly specified, the KF behaves better than the AF, but misspecification of Q leads to growing of the error in the KF. The AF is robust wrt the error in the specification of Q. This fact says in favor of the AF as an efficient tool for overcoming uncertainties in the ME.
Figure 5.
RMS of FE as a function of Q. The true value of Q is equal to 0. It is noted that the KF behaves better than the AF s.t. true Q but is more and more degraded as the ME becomes greater and greater. At the same time, the FE of the AF remains very robust.
6.2 Two dimensional system
6.2.1 Illustration of hypothesis HME
According to the notations in Section 5.3, consider the two-dimensional system (1) with Φ′=Φ′ij,Φ′11=1.02,Φ′12=0.1,Φ′21=0,Φ′22=0.9,H=11 with the true DME b′τ=col0.1,0.1′. Thus the first EiV is unstable, the second—stable [19].
Numerically one finds that the first SchVec is equal to u1=−1−7.0E−7T.
Figure 6 [19] shows the simulation results obtained on the basis of (37). One sees that, for na>10, the second component of wt is close to 0 whereas the first component becomes bigger and bigger (in absolute value) as na increases. Here, w′τ is a sequence of independent two-dimensional Gaussian random vectors of zero mean and variance 1. This means that the values of wt become more and more close to the subspace Ru1 spanned by u1, hence the HME is practically valid for na>10 in this example. Mention that, as a rule, in ocean numerical models, na is of order o(100) (na=800 or the MICOM model in the experiment in Section 6.3). See also [22].
Figure 6.
Two components of wt as functions of na.
6.2.2 Assimilation results
First simulation of the sequence of true system states x′τ,τ=1,…,390 has been carried out (see (35)). The observations are picked at τ=15,30,…,390. In terms of xt, the filtering problem then is of the form (1) and (2) s.t. t=τ15,Φ=Φ′15 (if no bias exists). When there is a bias, instead of wt stands ψt≔bt+wt.
In the experiment, the true system states x′t are generated by (35) s.t. b′τ=col0.1,0.1,w′t is zero mean with the covariance Q′=Id. The observation error is of zero mean and covariance R=0.16. For the state xt in the KF and NAF, the forecast is obtained at each assimilation instant t as x̂t+1/t=Φx̂t+b̂t. The simulation yields bt=0.22962.0589E−02 which results from applying (37) s.t. b′τ=col0.1,0.1.
Figure 7 depicts the time evolution of the KF and AF gains. One sees here as in the experiment with 1D system (Figure 2) that the KF gain is stabilized very quickly compared to that of the AF gain.
Figure 7.
Gain coefficients in KF and AF: The gains in KF and AF are identical at the beginning of the assimilation process.
Figure 8 (from [19]) shows the sample time average RMS of the state FE produced by the three filters NAF, KF, and AF. One sees that the AF outperforms the NAF and KF.
Figure 8.
Sample time average RMS of the state filtered error produced by the NAF, KF, and AF.
6.3 Data assimilation in the high-dimensional ocean model
To illustrate the effectiveness of the AF in dealing with uncertainties in HdSs, this section presents the results on data assimilation in the oceanic numerical model MICOM (Miami Isopycnic Coordinate Ocean Model) [19]. This MICOM describes the oceanic circulation in the North Atlantics. The model has four vertical layers with the state consisting of three variables x=huv where h is layer thickness, uv are two velocity components. The horizontal grid is 140×180. Totally at each time instant t we have the state xt of dimension 302400140×180×4layers×3variables. For more details on the configuration of this experiment, see [22].
The experiment is carried out on estimating the oceanic circulation using sea surface height (SSH) measurements. The SSH observation is available each 7 days (ds) (hence the observation window ΔT=7ds). Mention that simulating the circulation over 7 ds requires 800 model time steps δtintegration.
6.3.1 AF with optimal initial gain
First, in order to examine whether the method of optimal gain initialization, described in Section 3.2, is really useful for improving the filter performance, the optimization problem (21) has been solved. Symbolically, in the gain (20), Pr=IdQGTT where Id is the identity operator on the space of layer thickness h, QG is the quasi-geostrophy operator computing the correction for velocity using the SSH innovation ζ. The gain Ke computes the correction for h using ζ. The ECM M=Mv⊗Mh—Kronecker product of Mh—ECM of horizontal variable, Mv—ECM with vertical variable (see below for details). The two parameter matrices Θ and Λ are related to parameterization of Mv. The problem (21) is solved s.t. Θ=Id—identity operator.
The optimal parameters λi,i=1,…,4 are found by solving the minimization problem (21) using SPSA algorithm. Figure 9 shows the averaged values (see Comment 2.1) of λi,i=1,…,4 resulting during the optimization process. All λi,i=1,…,4 are initialized as λi=1,i=1,…,4.
Figure 9.
Control parameters λi,i=1,…,4 during optimization process.
The two NAFs are performed, one (denoted as NAFI) is with the gain (20) s.t. Θ=Id, Λ=Id, and the other (denoted by NAFOI) s.t. Θ=Id and λi,i=1,…,4 obtained by solving (21) (their values are those displayed at the end of the optimization process in Figure 9). The performances of these two NAFs are shown in Figure 10. One sees here that the NAFOI has improved considerably the quality of estimates of the velocity u-component compared with the NAFI. This result justifies that offline optimization (21) is an interesting strategy for finding the optimal initial gain in the NAF.
Figure 10.
Temporal average RMS of FE for total velocity u-component produced by the NAF s.t. the initial gain (red curve) and the NAF s.t. optimal initial gain resulting from solving (22) (green curve).
6.3.2 Estimating the ECM of ME
In practice, for real operational systems, information on the space of ME is not available or very poorly known. Usually, there is a big difference between the model and the real physical process and if the ME statistics are taken more or less properly, in some way, in the filtering algorithm, one can improve the filter performance and reduce the estimation error.
This idea is tested here by applying the HME in Section 4. We carry out the procedure for estimating the ECM of the ME by first constructing the subspace for the ME. For more details on the structure of the ECM M in the AF, see [23]. According to [23], the ECM M is assumed to be of the structure M=Mv⊗Mh—the Kronecker product of Mh with Mv where Mh is the ECM of the horizontal variable, Mv—ECM with vertical variable. Figure 11 displays RMS of FE for the u velocity component at the surface resulting from two AFs. The curve AF0U corresponds to the AF whose nonadaptive version has the gain computed on the basis of the ECM M using an ensemble of PE samples (generated by the PeSP in [18]). The curve AF3Ushows the performance of the AF with the modified ECM (by adding the vertical ME covariance Qv to the vertical ECM Mv). More precisely, Qv is assumed to belong to the subspace spanned by three leading EiVs of Mv. This choice is justified by the fact: the eigenvalue decomposition of Mv has the first three EiVs with the explained variances 67, 17, 15%, respectively. As the fourth EiVec has only the explained variance 0.7E-07%, it is dropped from the subspace constructed for the vertical ME. The better performance of the AF3U, in comparison with that of the AF0U, is apparently seen in Figure 11.
Figure 11.
Performance of the AF: (i) AF0U—no ME ECM has been taken into account; (ii) AF3U—with ME ECM computed in accordance with the HME.
The above experiment shows in details how, on the basis of HME, the subspace for the ME can be constructed, and how one estimates the ECM for the model error. The superior performance of the AF3U over that of the AF0U validates the usefulness of the HME which can serve as an important tool for estimating the ME and improving the performance of the AF for solving the data assimilation problems with HdSs.
7. Conclusions
One of the key assumptions to ensure the optimal performance of the KF is that a priori knowledge of the system model is given without any uncertainty. This assumption, however, is never valid in practice for dynamical systems under consideration. The uncertainties exist everywhere in modeling a real process like structural uncertainty, model parameterization, model resolution, model bias or ME statistics. For HdSs, order reduction introduced either in the original numerical model or in the filtering algorithms, inevitably leads to uncertainty in the ME, especially in geophysical numerical models.
Our focus in this chapter is to show how the AF solves efficiently filtering problems for systems operating in an uncertain environment.
As seen from this chapter, the AF has proven to be efficient to deal with uncertainties in the specification of the ME statistics, system bias or model reduction. The reasons of the success of the AF are that (i) it belongs to the class of parametrized stable filters; (ii) it is defined as the best member minimizing mean PE for the system outputs; (iii) The tuning parameters are chosen as elements of stabilizing gain and they are of no physical sense.
It is obvious from this chapter that the performance of the AF is comparable with that of the KF when perfect knowledge of all ME statistics is given, and it outperforms the KF in presence of uncertainties. This happens since the AF acquires knowledge during assimilation process, regardless of uncertainties existing in the filtering problems. From the computational point of view, implementation of the AF consumes much less memory and computational time than the KF or other assimilation methods.
Simple numerical examples and simulation results, presented in Sections 5 and 6, clearly demonstrate the advantages gained through application of the AF in dealing with uncertainties. These positive results encourage a wide application of the AF in different fields of technology and applied sciences like automatic control, finance, aerospace, space exploration, meteorology, and oceanography. A more in-depth and significant research on the capacity of the AF to deal with uncertainties is surely a challenge for the near future.
\n',keywords:"adaptive filter, innovation process, minimum prediction error, simultaneous perturbation stochastic approximation, filter stability",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/72778.pdf",chapterXML:"https://mts.intechopen.com/source/xml/72778.xml",downloadPdfUrl:"/chapter/pdf-download/72778",previewPdfUrl:"/chapter/pdf-preview/72778",totalDownloads:131,totalViews:0,totalCrossrefCites:0,dateSubmitted:"November 27th 2019",dateReviewed:"March 19th 2020",datePrePublished:"July 10th 2020",datePublished:"October 28th 2020",dateFinished:null,readingETA:"0",abstract:"In this contribution, the problem of data assimilation as state estimation for dynamical systems under uncertainties is addressed. This emphasize is put on high-dimensional systems context. Major difficulties in the design of data assimilation algorithms is a concern for computational resources (computational power and memory) and uncertainties (system parameters, statistics of model, and observational errors). The idea of the adaptive filter will be given in detail to see how it is possible to overcome uncertainties as well as to explain the main principle and tools for implementation of the adaptive filter for complex dynamical systems. Simple numerical examples are given to illustrate the principal differences of the AF with the Kalman filter and other methods. The simulation results are presented to compare the performance of the adaptive filter with the Kalman filter.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/72778",risUrl:"/chapter/ris/72778",signatures:"Hong Son Hoang and Remy Baraille",book:{id:"9966",title:"Dynamic Data Assimilation",subtitle:"Beating the Uncertainties",fullTitle:"Dynamic Data Assimilation - Beating the Uncertainties",slug:"dynamic-data-assimilation-beating-the-uncertainties",publishedDate:"October 28th 2020",bookSignature:"Dinesh G. Harkut",coverURL:"https://cdn.intechopen.com/books/images_new/9966.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"216122",title:"Dr.",name:"Dinesh G.",middleName:null,surname:"Harkut",slug:"dinesh-g.-harkut",fullName:"Dinesh G. Harkut"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"188919",title:"Dr.",name:"Hong Son",middleName:null,surname:"Hoang",fullName:"Hong Son Hoang",slug:"hong-son-hoang",email:"hhoang@shom.fr",position:null,institution:{name:"Naval Hydrographic and Oceanographic Service",institutionURL:null,country:{name:"France"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Adaptive filter",level:"1"},{id:"sec_3",title:"3. Covariance uncertainties and AF",level:"1"},{id:"sec_3_2",title:"3.1 Covariance uncertainties",level:"2"},{id:"sec_3_3",title:"3.1.1 Adjoint approach",level:"3"},{id:"sec_4_3",title:"3.1.2 Simultaneous perturbation stochastic approximation (SPSA) approach",level:"3"},{id:"sec_6_2",title:"3.2 Stability of the AF",level:"2"},{id:"sec_7_2",title:"3.3 On improving the initial gain",level:"2"},{id:"sec_9",title:"4. Joint estimation of state and model error in AF",level:"1"},{id:"sec_10",title:"5. Simple numerical examples",level:"1"},{id:"sec_10_2",title:"5.1 One-dimensional system",level:"2"},{id:"sec_11_2",title:"5.2 Two-dimensional system: specification of covariances",level:"2"},{id:"sec_11_3",title:"5.2.1 Stable filter",level:"3"},{id:"sec_12_3",title:"5.2.2 Unstable filter",level:"3"},{id:"sec_14_2",title:"5.3 Two-dimensional system: estimation of ME",level:"2"},{id:"sec_16",title:"6. Simulation results",level:"1"},{id:"sec_16_2",title:"6.1 One-dimensional system",level:"2"},{id:"sec_17_2",title:"6.2 Two dimensional system",level:"2"},{id:"sec_17_3",title:"6.2.1 Illustration of hypothesis HME",level:"3"},{id:"sec_18_3",title:"6.2.2 Assimilation results",level:"3"},{id:"sec_20_2",title:"6.3 Data assimilation in the high-dimensional ocean model",level:"2"},{id:"sec_20_3",title:"6.3.1 AF with optimal initial gain",level:"3"},{id:"sec_21_3",title:"6.3.2 Estimating the ECM of ME",level:"3"},{id:"sec_24",title:"7. Conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'Kalman REA. New approach to linear filtering and prediction problems. Journal of Basic Engineering. 1960;82:35-45. DOI: 10.1115/1.3662552'},{id:"B2",body:'Kailath T, Sayed AH, Hassibi B. Linear Estimation. NJ, Upper Saddle River: Prentice-Hall; 2000'},{id:"B3",body:'Liptser RS, Shiryaev AN. Statistics of Random Processes—I. General Theory. Berlin and Heidelberg: Springer-Verlag; 2001'},{id:"B4",body:'Sayed AH. Fundamentals of Adaptive Filtering. NJ: Wiley; 2003'},{id:"B5",body:'Kucera V. The discrete Riccati equation of optimal control. Kybernetika. 1972;8(5):430-447'},{id:"B6",body:'Hoang HS, Talagrand O, Baraille R. On the design of a stable adaptive filter for high dimensional systems. Automatica. 2001;37:341-359'},{id:"B7",body:'Simon D. Optimal State Estimation. Hoboken, NJ: John Wiley and Sons; 2006. ISBN: 978-0-471-70858-2'},{id:"B8",body:'Gustafsson F, Hendeby G. Some relations between extended and unscented Kalman filters. IEEE Transactions on Signal Processing. 2012;60(2):545-555'},{id:"B9",body:'Evensen G. The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dynamics. 2003;53(4):343-367. DOI: 10.1007/s10236-003-0036-9'},{id:"B10",body:'Chen Y, Snyder C. Assimilating vortex position with an ensemble Kalman filter. Monthly Weather Review. 2007;135(5):1828-1845. DOI: 10.1175/MWR3351.1'},{id:"B11",body:'Del Moral P. Non linear filtering: Interacting particle solution. Markov Processes and Related Fields. 1996;2(4):555-580'},{id:"B12",body:'Fitzgerald R. Divergence of the Kalman filter. IEEE Transactions on Automatic Control. 1971;16(6):736-747'},{id:"B13",body:'Hoang HS, De Mey P, Talagrand O, Baraille R. A new reduced-order adaptive filter for state estimation in high dimensional systems. Automatica. 1997;33(8):1475-1498'},{id:"B14",body:'Polyak BT. New method of stochastic approximation type. Automation and Remote Control. 1990;51(7):937-946'},{id:"B15",body:'Spall JC. Introduction to Stochastic Search and Optimization. New Jersey: Wiley; 2003. ISBN 978-0-471-33052-3'},{id:"B16",body:'Hoang HS, Baraille R. Stochastic simultaneous perturbation as powerful method for state and parameter estimation in high dimensional systems. In: Baswell AR, editor. Advances in Mathematics Research. Nova Science Publishers; 2015;20:117-148. ISBN: 978-1-63482-741-6c'},{id:"B17",body:'Hoang HS, Baraille R. A comparison study on performance of an adaptive filter with other estimation methods for state estimation in high-dimensional system. In: Hokimoto T, editor. Advances in Statistical Methodologies and their Application to Real Problems. Rijeka, Croatia: IntechOpen; 2017. pp. 29-52. DOI: 10.5772/67005'},{id:"B18",body:'Hoang HS, Baraille R. Prediction error sampling procedure based on dominant Schur decomposition. Application to state estimation in high dimensional oceanic model. Journal of Applied Mathematics and Computing. 2011;218(7):3689-3709'},{id:"B19",body:'Hoang HS, Baraille R. On estimation of model error by an adaptive filter. WSEAS Transactions on Systems and Control. 2019;14:158-168'},{id:"B20",body:'B. Friedland B. Treatment of bias in recursive filtering. IEEE Transactions on Automatic Control. 1969;14:359-367'},{id:"B21",body:'Albert AE. Regression and the Moore-Penrose Pseudo-Inverse. London: AP; 1972'},{id:"B22",body:'Hoang HS, Baraille R, Talagrand O. On an adaptive filter for altimetric data assimilation and its application to a primitive equation model, MICOM. Tellus 57A; 2005:153-170. DOI: 10.1111/j.1600-0870.2005.00094.x'},{id:"B23",body:'Hoang HS, Baraille R. On the efficient low cost procedure for estimation of high-dimensional prediction error covariance matrices. Automatica. 2017;83:317-330. DOI: 10.1016/j.automatica.2017.06.018'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Hong Son Hoang",address:"hhoang@shom.fr",affiliation:'
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Jean completed his primary and secondary school in Gabon. After high school, Jean started his undergraduate education at the University of Science and Technology of Masuku in Gabon where he obtained his Associate Degree in Biology and Chemistry. After obtaining a scholarship to study abroad, Jean pursued his higher education journey at the University of Rennes 1 in France where he obtained his bachelor’s degree and his master’s degree in Cell Biology and Physiology. In November 1999, Jean started his PhD work on the treatment of tuberculosis at the Pasteur Institute of Lille (France) while enrolled in the PhD program at the University of Lille 1. Jean defended his PhD in October 2003. Soon after, the same month, he left Europe for the United States of America where he accepted a position as post-doctoral fellow at the Public Health Research Institute in Newark, New Jersey. After 4 years of post-doc, Jean decided to Join the world of higher education by accepting a position as a Visiting Assistant Professor of Biology at Brigham Young University Hawaii. At the completion of his contract, Jean Joined Kent State University where he teaches a wide varieties of biological science courses. In addition to teaching, Jean focuses his research on molecular mechanisms of microbial pathogenesis and environmental microbiology. More recently, he expanded his research interest to epidemiology and biostatistics of noninfectious diseases, including epilepsy. Jean is a tenured faculty and an Associate Professor in the Department of Biological Sciences at Kent State University. He is the author of many peer-reviewed original research articles, reviews, and a book chapter.",institutionString:"Kent State University",institution:{name:"Kent State University",institutionURL:null,country:{name:"United States of America"}}},{id:"181260",title:"Dr.",name:"Taras",surname:"Kavetskyy",slug:"taras-kavetskyy",fullName:"Taras Kavetskyy",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Lublin University of Technology",institutionURL:null,country:{name:"Poland"}}},{id:"181414",title:"Dr.",name:"Maria Laura",surname:"Azcarate",slug:"maria-laura-azcarate",fullName:"Maria Laura Azcarate",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"183556",title:"Dr.",name:"Roberto",surname:"Uribe",slug:"roberto-uribe",fullName:"Roberto Uribe",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null}]},generic:{page:{slug:"our-story",title:"Our story",intro:"
The company was founded in Vienna in 2004 by Alex Lazinica and Vedran Kordic, two PhD students researching robotics. While completing our PhDs, we found it difficult to access the research we needed. So, we decided to create a new Open Access publisher. A better one, where researchers like us could find the information they needed easily. The result is IntechOpen, an Open Access publisher that puts the academic needs of the researchers before the business interests of publishers.
",metaTitle:"Our story",metaDescription:"The company was founded in Vienna in 2004 by Alex Lazinica and Vedran Kordic, two PhD students researching robotics. While completing our PhDs, we found it difficult to access the research we needed. So, we decided to create a new Open Access publisher. A better one, where researchers like us could find the information they needed easily. The result is IntechOpen, an Open Access publisher that puts the academic needs of the researchers before the business interests of publishers.",metaKeywords:null,canonicalURL:"/page/our-story",contentRaw:'[{"type":"htmlEditorComponent","content":"
We started by publishing journals and books from the fields of science we were most familiar with - AI, robotics, manufacturing and operations research. Through our growing network of institutions and authors, we soon expanded into related fields like environmental engineering, nanotechnology, computer science, renewable energy and electrical engineering, Today, we are the world’s largest Open Access publisher of scientific research, with over 4,200 books and 54,000 scientific works including peer-reviewed content from more than 116,000 scientists spanning 161 countries. Our authors range from globally-renowned Nobel Prize winners to up-and-coming researchers at the cutting edge of scientific discovery.
\\n\\n
In the same year that IntechOpen was founded, we launched what was at the time the first ever Open Access, peer-reviewed journal in its field: the International Journal of Advanced Robotic Systems (IJARS).
\\n\\n
The IntechOpen timeline
\\n\\n
2004
\\n\\n
\\n\\t
Intech Open is founded in Vienna, Austria, by Alex Lazinica and Vedran Kordic, two PhD students, and their first Open Access journals and books are published.
\\n\\t
Alex and Vedran launch the first Open Access, peer-reviewed robotics journal and IntechOpen’s flagship publication, the International Journal of Advanced Robotic Systems (IJARS).
\\n
\\n\\n
2005
\\n\\n
\\n\\t
IntechOpen publishes its first Open Access book: Cutting Edge Robotics.
\\n
\\n\\n
2006
\\n\\n
\\n\\t
IntechOpen publishes a special issue of IJARS, featuring contributions from NASA scientists regarding the Mars Exploration Rover missions.
\\n
\\n\\n
2008
\\n\\n
\\n\\t
Downloads milestone: 200,000 downloads reached
\\n
\\n\\n
2009
\\n\\n
\\n\\t
Publishing milestone: the first 100 Open Access STM books are published
\\n
\\n\\n
2010
\\n\\n
\\n\\t
Downloads milestone: one million downloads reached
\\n\\t
IntechOpen expands its book publishing into a new field: medicine.
\\n
\\n\\n
2011
\\n\\n
\\n\\t
Publishing milestone: More than five million downloads reached
\\n\\t
IntechOpen publishes 1996 Nobel Prize in Chemistry winner Harold W. Kroto’s “Strategies to Successfully Cross-Link Carbon Nanotubes”. Find it here.
\\n\\t
IntechOpen and TBI collaborate on a project to explore the changing needs of researchers and the evolving ways that they discover, publish and exchange information. The result is the survey “Author Attitudes Towards Open Access Publishing: A Market Research Program”.
\\n\\t
IntechOpen hosts SHOW - Share Open Access Worldwide; a series of lectures, debates, round-tables and events to bring people together in discussion of open source principles, intellectual property, content licensing innovations, remixed and shared culture and free knowledge.
\\n
\\n\\n
2012
\\n\\n
\\n\\t
Publishing milestone: 10 million downloads reached
\\n\\t
IntechOpen holds Interact2012, a free series of workshops held by figureheads of the scientific community including Professor Hiroshi Ishiguro, director of the Intelligent Robotics Laboratory, who took the audience through some of the most impressive human-robot interactions observed in his lab.
\\n
\\n\\n
2013
\\n\\n
\\n\\t
IntechOpen joins the Committee on Publication Ethics (COPE) as part of a commitment to guaranteeing the highest standards of publishing.
\\n
\\n\\n
2014
\\n\\n
\\n\\t
IntechOpen turns 10, with more than 30 million downloads to date.
\\n\\t
IntechOpen appoints its first Regional Representatives - members of the team situated around the world dedicated to increasing the visibility of our authors’ published work within their local scientific communities.
\\n
\\n\\n
2015
\\n\\n
\\n\\t
Downloads milestone: More than 70 million downloads reached, more than doubling since the previous year.
\\n\\t
Publishing milestone: IntechOpen publishes its 2,500th book and 40,000th Open Access chapter, reaching 20,000 citations in Thomson Reuters ISI Web of Science.
\\n\\t
40 IntechOpen authors are included in the top one per cent of the world’s most-cited researchers.
\\n\\t
Thomson Reuters’ ISI Web of Science Book Citation Index begins indexing IntechOpen’s books in its database.
\\n
\\n\\n
2016
\\n\\n
\\n\\t
IntechOpen is identified as a world leader in Simba Information’s Open Access Book Publishing 2016-2020 report and forecast. IntechOpen came in as the world’s largest Open Access book publisher by title count.
\\n
\\n\\n
2017
\\n\\n
\\n\\t
Downloads milestone: IntechOpen reaches more than 100 million downloads
\\n\\t
Publishing milestone: IntechOpen publishes its 3,000th Open Access book, making it the largest Open Access book collection in the world
We started by publishing journals and books from the fields of science we were most familiar with - AI, robotics, manufacturing and operations research. Through our growing network of institutions and authors, we soon expanded into related fields like environmental engineering, nanotechnology, computer science, renewable energy and electrical engineering, Today, we are the world’s largest Open Access publisher of scientific research, with over 4,200 books and 54,000 scientific works including peer-reviewed content from more than 116,000 scientists spanning 161 countries. Our authors range from globally-renowned Nobel Prize winners to up-and-coming researchers at the cutting edge of scientific discovery.
\n\n
In the same year that IntechOpen was founded, we launched what was at the time the first ever Open Access, peer-reviewed journal in its field: the International Journal of Advanced Robotic Systems (IJARS).
\n\n
The IntechOpen timeline
\n\n
2004
\n\n
\n\t
Intech Open is founded in Vienna, Austria, by Alex Lazinica and Vedran Kordic, two PhD students, and their first Open Access journals and books are published.
\n\t
Alex and Vedran launch the first Open Access, peer-reviewed robotics journal and IntechOpen’s flagship publication, the International Journal of Advanced Robotic Systems (IJARS).
\n
\n\n
2005
\n\n
\n\t
IntechOpen publishes its first Open Access book: Cutting Edge Robotics.
\n
\n\n
2006
\n\n
\n\t
IntechOpen publishes a special issue of IJARS, featuring contributions from NASA scientists regarding the Mars Exploration Rover missions.
\n
\n\n
2008
\n\n
\n\t
Downloads milestone: 200,000 downloads reached
\n
\n\n
2009
\n\n
\n\t
Publishing milestone: the first 100 Open Access STM books are published
\n
\n\n
2010
\n\n
\n\t
Downloads milestone: one million downloads reached
\n\t
IntechOpen expands its book publishing into a new field: medicine.
\n
\n\n
2011
\n\n
\n\t
Publishing milestone: More than five million downloads reached
\n\t
IntechOpen publishes 1996 Nobel Prize in Chemistry winner Harold W. Kroto’s “Strategies to Successfully Cross-Link Carbon Nanotubes”. Find it here.
\n\t
IntechOpen and TBI collaborate on a project to explore the changing needs of researchers and the evolving ways that they discover, publish and exchange information. The result is the survey “Author Attitudes Towards Open Access Publishing: A Market Research Program”.
\n\t
IntechOpen hosts SHOW - Share Open Access Worldwide; a series of lectures, debates, round-tables and events to bring people together in discussion of open source principles, intellectual property, content licensing innovations, remixed and shared culture and free knowledge.
\n
\n\n
2012
\n\n
\n\t
Publishing milestone: 10 million downloads reached
\n\t
IntechOpen holds Interact2012, a free series of workshops held by figureheads of the scientific community including Professor Hiroshi Ishiguro, director of the Intelligent Robotics Laboratory, who took the audience through some of the most impressive human-robot interactions observed in his lab.
\n
\n\n
2013
\n\n
\n\t
IntechOpen joins the Committee on Publication Ethics (COPE) as part of a commitment to guaranteeing the highest standards of publishing.
\n
\n\n
2014
\n\n
\n\t
IntechOpen turns 10, with more than 30 million downloads to date.
\n\t
IntechOpen appoints its first Regional Representatives - members of the team situated around the world dedicated to increasing the visibility of our authors’ published work within their local scientific communities.
\n
\n\n
2015
\n\n
\n\t
Downloads milestone: More than 70 million downloads reached, more than doubling since the previous year.
\n\t
Publishing milestone: IntechOpen publishes its 2,500th book and 40,000th Open Access chapter, reaching 20,000 citations in Thomson Reuters ISI Web of Science.
\n\t
40 IntechOpen authors are included in the top one per cent of the world’s most-cited researchers.
\n\t
Thomson Reuters’ ISI Web of Science Book Citation Index begins indexing IntechOpen’s books in its database.
\n
\n\n
2016
\n\n
\n\t
IntechOpen is identified as a world leader in Simba Information’s Open Access Book Publishing 2016-2020 report and forecast. IntechOpen came in as the world’s largest Open Access book publisher by title count.
\n
\n\n
2017
\n\n
\n\t
Downloads milestone: IntechOpen reaches more than 100 million downloads
\n\t
Publishing milestone: IntechOpen publishes its 3,000th Open Access book, making it the largest Open Access book collection in the world
\n
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I am also a member of the team in charge for the supervision of Ph.D. students in the fields of development of silicon based planar waveguide sensor devices, study of inelastic electron tunnelling in planar tunnelling nanostructures for sensing applications and development of organotellurium(IV) compounds for semiconductor applications. I am a specialist in data analysis techniques and nanosurface structure. I have served as the editor for many books, been a member of the editorial board in science journals, have published many papers and hold many patents.",institutionString:null,institution:{name:"Sheffield Hallam University",country:{name:"United Kingdom"}}},{id:"54525",title:"Prof.",name:"Abdul Latif",middleName:null,surname:"Ahmad",slug:"abdul-latif-ahmad",fullName:"Abdul Latif Ahmad",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"20567",title:"Prof.",name:"Ado",middleName:null,surname:"Jorio",slug:"ado-jorio",fullName:"Ado Jorio",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Universidade Federal de Minas Gerais",country:{name:"Brazil"}}},{id:"47940",title:"Dr.",name:"Alberto",middleName:null,surname:"Mantovani",slug:"alberto-mantovani",fullName:"Alberto Mantovani",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"12392",title:"Mr.",name:"Alex",middleName:null,surname:"Lazinica",slug:"alex-lazinica",fullName:"Alex Lazinica",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/12392/images/7282_n.png",biography:"Alex Lazinica is the founder and CEO of IntechOpen. After obtaining a Master's degree in Mechanical Engineering, he continued his PhD studies in Robotics at the Vienna University of Technology. Here he worked as a robotic researcher with the university's Intelligent Manufacturing Systems Group as well as a guest researcher at various European universities, including the Swiss Federal Institute of Technology Lausanne (EPFL). During this time he published more than 20 scientific papers, gave presentations, served as a reviewer for major robotic journals and conferences and most importantly he co-founded and built the International Journal of Advanced Robotic Systems- world's first Open Access journal in the field of robotics. Starting this journal was a pivotal point in his career, since it was a pathway to founding IntechOpen - Open Access publisher focused on addressing academic researchers needs. Alex is a personification of IntechOpen key values being trusted, open and entrepreneurial. Today his focus is on defining the growth and development strategy for the company.",institutionString:null,institution:{name:"TU Wien",country:{name:"Austria"}}},{id:"19816",title:"Prof.",name:"Alexander",middleName:null,surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/19816/images/1607_n.jpg",biography:"Alexander I. Kokorin: born: 1947, Moscow; DSc., PhD; Principal Research Fellow (Research Professor) of Department of Kinetics and Catalysis, N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow.\r\nArea of research interests: physical chemistry of complex-organized molecular and nanosized systems, including polymer-metal complexes; the surface of doped oxide semiconductors. 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