A New Deterministic Ensemble Kalman Filter with One-Step-Ahead Smoothing for Storm Surge Forecasting

A new deterministic Ensemble Kalman Filter with one-step-ahead smoothing for storm surge forecasting

Thesis by

Naila Raboudi

In Partial Fulﬁllment of the Requirements

For the Degree of

Masters of Science

King Abdullah University of Science and Technology

Thuwal, Kingdom of Saudi Arabia

November, 2016 2

EXAMINATION COMMITTEE PAGE

The thesis of Naila Raboudi is approved by the examination committee

Committee Chairperson: Ibrahim Hoteit Committee Members: Omar Knio, Shuyu Sun, Boujemaa Ait-El-Fquih 3

©November, 2016 Naila Raboudi

ABSTRACT

A new deterministic Ensemble Kalman Filter with one-step-ahead smoothing for storm surge forecasting Naila Raboudi

The Ensemble Kalman Filter (EnKF) is a popular data assimilation method for state- parameter estimation. Following a sequential assimilation strategy, it breaks the problem into alternating cycles of forecast and analysis steps. In the forecast step, the dynamical model is used to integrate a stochastic sample approximating the state analysis distribution (called analysis ensemble) to obtain a forecast ensemble. In the analysis step, the forecast ensemble is updated with the incoming observation using a Kalman-like correction, which is then used for the next forecast step. In realistic large-scale applications, EnKFs are implemented with limited ensembles, and often poorly known model errors statistics, leading to a crude approximation of the forecast covariance. This strongly limits the filter performance. Recently, a new EnKF was proposed in [1] following a one-step-ahead smoothing strategy (EnKF-OSA), which involves an OSA smoothing of the state between two successive analysis. At each time step, EnKF-OSA exploits the observation twice. The incoming observation is first used to smooth the ensemble at the previous time step. The resulting smoothed ensemble is then integrated forward to compute a ”pseudo forecast” ensemble, which is again updated with the same observation. The idea of constraining the state with future observations is to add more information in the estimation process in order to mitigate for the sub-optimal character of EnKF-like methods. The second EnKF-OSA ”forecast” is computed from the smoothed ensemble and should therefore provide an improved background. 5 In this work, we propose a deterministic variant of the EnKF-OSA, based on the Sin- gular Evolutive Interpolated Ensemble Kalman (SEIK) filter. The motivation behind this is to avoid the observations perturbations of the EnKF in order to improve the scheme’s behavior when assimilating big data sets with small ensembles. The new SEIK-OSA scheme is implemented and its efficiency is demonstrated by performing assimilation experiments with the highly nonlinear Lorenz model and a realistic setting of the Advanced Circulation (ADCIRC) model configured for storm surge forecasting in the Gulf of Mexico during Hurricane Ike.

Key words : Data assimilation, Kalman filter, Ensemble Kalman filter, Singular Evolutive Interpolated Ensemble Kalman filter, Smoothing-based filtering. 6

ACKNOWLEDGEMENTS

Because I owe a particular debt of respect to all those who have listened, advised and criticized me, I would like to hereby express them, from my heart, the deepest gratitude through these short lines. Many people assisted me in one way or an other with the elaboration of this work. This Master project is made under the sincere guidance of my advisor, Prof. Ibrahim Hoteit, to whom I would like to extend a special note of gratitude for his generous technical and moral support, for being always patient and willing to assist me with my work and for all the time and effort that he spent helping me. It is also my duty to express my special thanks and appreciations to Dr. Boujemaa Ait El Fquih who has willingly assisted me with his abilities, for all the time and effort he helped me out with and for his valuable guidance, endless support and understanding spirit. I would especially like to express my gratitude towards my amazing family for the love, support, and constant encouragement I have gotten over the years. I undoubtedly could not have done this without their kind co-operation and encouragement which helped me in the completion of this project. I am also indebted to KAUST for the education and the facilities that it is providing. I want also to acknowledge with much appreciation the crucial role of all my professors who educated me during my stay here. Last but not least, many thanks and gratitudes are extended to the highly esteemed members of the examination committee, Prof. Omar Knio and Prof. Shuyu Sun, who accepted to judge my work and who I wish hopefully to be satisfied. 7

TABLE OF CONTENTS

Examination Committee Page 2

Abstract 4

Acknowledgements 6

List of Figures 9

List of Tables 12

1 Introduction 13

2 Kalman Filtering 18 2.1 Introduction ...... 18 2.2 Bayesian ﬁltering ...... 18 2.3 The Kalman Filter (KF) ...... 21 2.4 The (stochastic) Ensemble Kalman ﬁlter (EnKF) ...... 25 2.5 EnKF limitations and and auxiliary methods ...... 27 2.5.1 EnKF limitations ...... 27 2.5.2 Methods to mitigate EnKF limitations ...... 29 2.6 Discussion ...... 34

3 Ensemble Kalman Filtering with One-Step-Ahead Smoothing 35 3.1 Introduction ...... 35 3.2 The Kalman ﬁltering with one-step-ahead smoothing (KF-OSA) . . . 35 3.2.1 Generic algorithm ...... 36 3.2.2 KF-OSA equations ...... 37 3.3 Ensemble formulation (EnKF-OSA) ...... 39 3.3.1 Smoothing step ...... 40 3.3.2 Analysis step ...... 41 3.4 Summary of the EnKF-OSA algorithm ...... 42 8 3.5 Discussion ...... 43

4 Deterministic Ensemble Kalman Filtering with One-Step-Ahead Smooth- ing 44 4.1 Introduction ...... 44 4.2 The Singular Evolutive Interpolated Kalman ﬁlter (SEIK) ...... 45 4.3 SEIK ﬁlter with One-Step-Ahead smoothing (SEIK-OSA) ...... 48 4.3.1 Smoothing step ...... 48 4.3.2 Analysis step ...... 51 4.4 Summary of the SEIK-OSA algorithms ...... 63 4.5 Conclusion ...... 65

5 Numerical Experiments & Results 66 5.1 Introduction ...... 66 5.2 Numerical experiments with the Lorenz-96 model ...... 66 5.2.1 Experimental setting ...... 66 5.2.2 Results and discussion ...... 69 5.3 Numerical experiments with a storm surge model ...... 88 5.3.1 An overview of ADCIRC model ...... 88 5.3.2 Experimental design and implementation ...... 89 5.3.3 Results and discussion ...... 96 5.4 Summary and conclusions ...... 99

6 Concluding Remarks 101

References 104

Appendices 108 9

LIST OF FIGURES

5.1 Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis). The two filters are implemented with 10 members and assimilation of observations from (top) all model variables (middle) half and (bottom) quarter of the variables at every 4 model time steps (or 24 h in real time). A logarithmic color scale is used to emphasize the low RMSE values. The minimum-averaged RM- SEs are indicated by asterisks, and their associated values are given in the title. White boxes indicate divergence of the filter...... 70 5.2 Same as Fig. (5.1), but for 20 ensemble members...... 71 5.3 Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis). The two filters are implemented with 20 members and assimilation of observations every (top) 1 model time step (middle) 2 and (bottom) 4 model time steps. A logarithmic color scale is used to emphasize the low RMSE values...... 71 5.4 Same as Fig. (5.1), but for 20 ensemble members and F = 6 (instead of 8) in filters...... 73 5.5 Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis). All filters are implemented with 10 members and assimilation of observations from (top) all model variables (middle) half and (bottom) quarter of the variables at every 4 model time steps (or 24 h in real time). A logarithmic color scale is used to emphasize the low RMSE values. The minimum-averaged RMSEs are indicated by asterisks, and their associated values are given in the title. White boxes indicate divergence of the filter...... 74 5.6 Same as Fig. (5.5), but for 20 ensemble members...... 75 5.7 Same as Fig. (5.5), but for 40 ensemble members...... 76 5.8 Minimum average RMSE for all tested filters (EnKF-Reg, EnKF-OSA, SEIK-Reg and SEIK-OSA) as a function of the ensemble size. left (All) middle (half) and right (quarter) observations are observed every 4 model time steps...... 77 10 5.9 Same as Fig. (5.5), but for 20 ensemble members and assimilation every model time step...... 79 5.10 Same as Fig. (5.5), but for 20 ensemble members and assimilation every two model time steps...... 80 5.11 Same as Fig. (5.5), but for 20 ensemble members and assimilation every six model time steps...... 81 5.12 Minimum average RMSE for all tested filters (EnKF-Reg, EnKF-OSA, SEIK-Reg and SEIK-OSA) as a function of the frequency of observations. (left) All (middle) half and (right) quarter observations are

observed every every 4 model steps with Ne=20...... 81 5.13 Same as Fig. (5.5), but for 20 ensemble members and observational error variance equal to 0.1...... 83 5.14 Same as Fig. (5.5), but for 20 ensemble members and observational error variance equal to 2...... 84 5.15 Minimum average RMSE for all tested ﬁlters (EnKF-Reg, EnKF-OSA, SEIK-Reg and SEIK-OSA) as a function of the measurement error variance. (left) All (middle) half and (right) quarter observations are

assimilated every 4 model steps with Ne=20...... 85 5.16 Same as Fig. (5.16), but for 20 ensemble members and assimilation every 4 model time steps. Simulations are carried out with an incorrect forcing F =6...... 85 5.17 Same as Fig. (5.16), but with assimilation every 2 model time steps. . 86 5.18 Track of Hurricane Ike through the Gulf of Mexico. The circles with annotations are the locations of landfall on Sept. 13, 2008 at 07:10 UTC and the locations of the hurricane approximately 48 and 72 h before landfall. The x axis represents the longitude and the y axis the latitude in degrees...... 89 5.19 Finite element mesh of the Gulf of Mexico. The x axis represents the longitude and the y axis the latitude in degrees...... 91 5.20 Distribution of the 43 observation stations. The x axis represents the longitude and the y axis the latitude in degrees...... 92 5.21 Coastal-averaged analysis RMSE with local analysis for different radii and inflation factors. The results are presented for an ensemble size of 10 with the hybrid SEIK and 5 members with the hybrid EIK-OSA for Hurricane Ike simulations. Minimum values are indicated in the title. 96 11 5.22 Plots of true and forecast states associated with the minimum of forecast RMSE at some stations located close to the landfall areas. . . . . 97 5.23 Time series of the coastal RMSE...... 98 5.24 Plots of free surface absolute elevation errors one hour before the landfall using the best empirically determined best choices of inflation factor and localization...... 99 12

LIST OF TABLES

5.1 Best percentages of improvement, in terms of RMSE, introduced by the OSA-based filters compared to the standard filters using the same ensemble size...... 77 5.2 Best percentages of improvement, in terms of RMSE, introduced by the OSA-based filters compared to the standard filters using only half of the ensemble size. The (-) sign means that the standard filter with

2Ne produces better estimates than the OSA-based ﬁlter with Ne... 78 13

Chapter 1

Introduction

The devastation due to storm surge flooding caused by extreme wind waves is a severe apprehension along the coastal regions. Storm surges are often considered as the greatest threat to life along the coasts. Large death tolls have, indeed, resulted from the rise of the ocean associated with many hurricanes that have made landfall. Hurricane Ike (2008) is a prime example of the damage and devastation that can be caused by a surge. It is, in fact, the most destructive hurricane to make landfall on the Texas coast since the Galveston Hurricane of 1900. It was estimated by some as the third costliest hurricane ever to make landfall in the United States. Ike caused not only incredible monetary damage, estimated at nearly 29.6 billion in the United States alone, but also nearly 200 deaths [2]. These devastating weather events have raised the need for increased forecast accuracy to reduce their impact. Providing accurate forecasts of storm surges, delivered in real time, is then a problem of critical importance as they could result in more timely evacuations and help decision mak- ers planning for mitigation before severe storms occur. Numerical ocean models are considered today as an essential tool to predict the sea level rise. Despite the tremen- dous progress in computing resources that enabled the use of high resolution models of storm surge forecasting, these remain subject to significant uncertainties from various sources that limit their performances. Sources of uncertainties include poorly known wind and model parameters, specification of physical properties, boundary conditions and initial conditions, most of which being out of reach of the available measurements. 14 Data assimilation optimally exploits available observations to enhance models performances. Conversely, the model provides a space-time dynamical interpolation of the data. This technique has found many applications mainly in atmosphere, ocean and hydrology. There are two classes of data assimilation methods: variational methods and sequential methods. The first category is based on the optimal control theory and aspires at optimizing an objective function measuring the misfit between observations and model trajectory over a given period of time through adjustments of a well chosen set of control parameters [3]. An optimization algorithm is then used to minimize the cost-function [4]. Three-dimensional variational data assimilation (3D- VAR) assumes that the forecast and the observations, have errors described by static error covariances [5]. These assumptions facilitate solving realistic problems computationally, however, 3DVAR misses the time-dependent error dynamics common in realistic systems. Four-dimensional variational data assimilation (4D-VAR) was then introduced as a more sophisticated scheme using an adjoint model to include the time dimension. Variational methods were shown to provide good results, but are difficult to implement with sophisticated models due to their prohibitive computational bur- den. In contrast, sequential methods use a probabilistic framework and give estimates of the whole system state sequentially by propagating information forward in time. They break the Bayesian estimation problem into cycles of alternating forecast and analysis steps. These methods have have seen widespread use in many geophysical applications owing to their reasonable computational cost and efficiency. The classical Kalman filter (KF) [6] is the most popular sequential data assimilation method, providing the optimal state estimate of a linear system under additive, state- independent Gaussian model and observation errors. At times where no observations are available, the dynamical model is used to integrate an earlier state estimate forward in time, which is known as the forecast step. In the analysis step, incoming observations are used to update the forecast state. The updated state estimate is called 15 the analysis and is used as the starting point for the next forecast. Geophysical models are, however, nonlinear and thus, alternatives to the KF are needed. The extended Kalman filter (EKF) is an early attempt to adapt the KF to nonlinear problems by first linearizing the model around the most recent state estimate and then applying the KF on the resulting linearized system. Applying EKF to large dimensional nonlinear systems is not possible because of the prohibitive cost of manipulating the covariance matrices of the state errors estimates that exceeds the capacities of current computer resources. The Ensemble Kalman Filter (EnKF), originally introduced by [7] to handle large nonlinear oceanic models, is a Monte Carlo approximation of the Kalman filter. Its key idea is to represent the distribution of the system state by a collection of realizations, called an ensemble. The state estimate is then estimated as the mean of the ensemble and the associated error covariance matrix as the sample covariance of the ensemble. In the forecast step, the analysis ensemble is integrated forward in time using the model which gives rise to a forecast ensemble, which is, in turn, updated with the incoming observation using a Kalman-like correction during the analysis step. The resulting ensemble will serve as a starting point for the next forecast step. Preserving enough ensemble spread was generally found to be beneficial for an EnKF behavior [8]. [9] suggested to perturb the observations with random noise sampled from the observational error covariance matrix before using the observation to update each member of the ensemble so that the EnKF error covariance asymptotically matches that of the KF. Later, this became to be known as the stochastic EnKF. This approach introduces, however, an additional source of sampling error and often underestimates the analysis error covariance especially when the filter is implemented with small ensembles [8] [10]. An alternative deterministic formulation of the EnKF was then introduced [11], in which the update step of the forecast ensemble mean and covariance are performed 16 exactly as in the KF. A ”resampling” step was then needed to generate a new ensemble matching the analysis state and its error covariance. This is the idea behind the Singular Evolutive Interpolated Kalman (SEIK) filter which also uses an orthog- onal random transformation [12]. In the Ensemble Transform Kalman filter (ETKF) [13], the Ensemble Adjustment Kalman filter (EAKF) [14], the Ensemble Square-root Kalman filter with sequential processing of observations (EnSRF) [8], the resampling is implicitly implemented through some kind of square-root transformation of the ensemble. This contrasts with the random implicit resampling of the stochastic EnKF (via the observations perturbations) which, in fact, samples the ensemble from a Gaussian distribution with the KF estimate and error covariance as the first two moments. [10] In realistic large-scale applications, EnKFs are usually implemented with restricted ensemble sizes relatively to the dimension of the state space. Small ensembles ensure reasonable computational cost especially in operational applications where timely forecasts are required, however, may lead to a crude approximation of the forecast covariance. The poor knowledge of model errors has also a similar impact on the filter behavior. The associated errors are likely to propagate to subsequent steps of the filter which strongly limits its performance. Recently, [1] introduced a new stochastic EnKF algorithm following the One-Step-Ahead smoothing (OSA-based) formulation of the Bayesian filtering problem for state-parameter estimation. Compared to the standard formulation of the EnKF, the proposed stochastic (EnKF-OSA) offers the key advantage of exploiting the same observation twice, once for state smoothing and another for updating the forecasted smoothed ensemble within a fully consistent Bayesian formulation of the state estimation problem. Computationally, this means that the EnKF-OSA is twice more expensive than the standard EnKF. However, the information provided by the observation is exploited twice which is expected to improve the background leading to more accurate estimates. The method was tested 17 with a groundwater flow model to estimate the hydraulic head and the conductivity parameter field and was shown to be more beneficial in terms of estimation accuracy. In this work, we first assess the relevance of EnKF-OSA for state estimation in comparison to the standard EnKF. Then, we propose two deterministic variants of the EnKF-OSA based on the SEIK filter and compare their performances to the stochastic EnKF-OSA. The goal is to develop an EnKF-OSA algorithm that avoids the observations perturbations to improve the scheme’s behavior when assimilating big data sets with small ensembles which is expected to make it suitable for storm surge forecasting in particular and realistic oceanic and atmospheric data assimilation problems in general. The algorithms are first tested with the strongly nonlinear Lorenz model through an extensive set of sensitivity experiments to study their responses to different scenarios before applying the SEIK-OSA scheme to a realistic setting of the ADCIRC model configured for storm surge forecasting in the Gulf of Mexico during Hurricane Ike. This thesis is organized as follows. Chapter 1 recalls the concept of data assimilation, focusing on the Kalman filter. The stochastic and deterministic ensemble formula- tions of the Kalman filter and theit limitations are then presented and discussed. Chapter 2 derives the stochastic EnKF-OSA algorithm and Chapter 3 introduces two new deterministic variants of the EnKF-OSA based on the SEIK filter. Chapter 4 presents the experimental setting with the Lorenz-96 model and the storm surge forecasting model ADCIRC and discusses the numerical results. 18

Chapter 2

Kalman Filtering

2.1 Introduction

Data assimilation (DA) is the process by which observations are combined with prior knowledge to estimate as accurately as possible the true state of a dynamical system. The Bayesian estimation framework provides a coherent probabilistic approach for DA. In the linear Gaussian case, the Kalman ﬁlter (KF) provides the optimal solution of the Bayesian estimation problem. Nonlinearities and prohibitive computational cost in realistic applications led to the introduction of ensemble based KFs (EnKFs), which are Monte Carlo implementations of the KF designed for large scale nonlinear systems [15]. In this chapter, we will start by presenting the Bayesian estimation problem, from which we will then derive the KF. The EnKF algorithm as well as its limitations will be then introduced with some state of the art auxiliary methods to enhance the ﬁlter performances in realistic large scale applications.

2.2 Bayesian ﬁltering

We explore DA from a Bayesian estimation perspective. Epstein [16] was among the ﬁrst to consider the Bayesian methods in atmospheric sciences. Lorenc [17] then presented a comprehensive Bayesian perspective for data assimilation. DA problems are generally interested in estimating a dynamical process x = {x } ∈ Nx given n n∈N R a set of observations y = {y } ∈ No . The prior distribution, p(x), quanties n n∈N R the a priori understanding of the quantity of interest. The posterior distribution, 19 p(x|y), deﬁnes the distribution of the same quantity given the observed data. This distribution can be then seen as the update of the prior knowledge as summarized

def in p(x) given the actual observation y. Throughout the thesis, we deﬁne y0:n =

{y0, y1, ··· , yn}, p(xn) and p(xn|y0:p) are the prior probability density function (pdf) of xn and the posterior pdf of xn given y0:p, respectively. All other used pdfs are deﬁned in a similar way. A classical estimation problem is described by the following well-known state-space system.   xn = Mn−1(xn−1) + ηn−1 , (2.1)  yn = Hn(xn) + εn where,

Nx • xn ∈ R is the system state at time step n of dimension Nx.

No • yn ∈ R is the corresponding observation of xn.

Nx • ηn ∈ R is the state model noise.

No • εn ∈ R is the observation noise.

• Mn−1 is the dynamical model that integrates the model state forward in time

from xn−1 to xn.

• Hn is an observation operator that projects xn from the state space onto the observation space.

The state transition function, Mn−1 and the observation operator Hn are not neces- sarily linear. The model noise, η = {η } , and the observation noise, ε = {ε } , n n∈N n n∈N are assumed to be independent, jointly independent and independent of the initial state x0. Using the Bayes rule, the posterior distribution of the states given the data is given 20 by:

p(x0:n|y0:n) ∝ p(y0:n|x0:n)p(x0:n). (2.2)

Typically, these problems are addressed using the Markov assumption. That is, the state at any time tn depends only on the state at the previous time tn−1.

n Y p(x0:n) = p(x0) p(xk|xk−1), (2.3) k=1

where p(xk|xk−1) is the evolution distribution and p(x0) is the distribution of the initial state. The observations, knowing the true state, are also assumed to be independent.

n Y p(y0:n|x0:n) = p(x0) p(yk|xk). (2.4) k=0

Using (2.3) and (2.5), the Bayes rule in (2.2) can be rewritten as:

n Y p(x0:n|y0:n) ∝ p(x0) p(yk|xk)p(xk|xk−1). (2.5) k=1

Writing the Bayes rule under this form, the update appears to be sequential in the sense that, once a new observation is available, the previous optimal state estimate could be updated using this data.

In a ﬁltering problem, the density p(xn−1|y0:n−1) is assumed to be available and is then used to compute:

– The forecast distribution p(xn|y0:n−1) from the Markov assumption as:

Z p(xn|y0:n−1) = p(xn|xn−1)p(xn−1|y0:n−1)dxn−1. (2.6) 21

– The analysis distribution p(xn|yn−1) from the Bayes rule:

p(xn|y0:n) = p(xn|y0:n−1),

∝ p(yn|xn, y0:n−1)p(xn|y0:n−1),

= p(yn|xn)p(xn|y0:n−1). (2.7)

Determining an analytical solution of the Bayesian ﬁltering problem is rarely possible. It is however possible to obtain analytical representations for the forecast and analysis distributions (2.6)-(2.7). when the evolution and observation operators (Mn−1 and

Hn in (2.1)) are linear and the errors and the initial state x0 are Gaussian. In this case, the Kalman ﬁlter (KF) is the optimal algorithmic solution to the Bayesian ﬁltering problem. [6].

2.3 The Kalman Filter (KF)

In a KF setting, the posterior distribution is always Gaussian, and thus, could be fully characterized by the mean and the covariance [6]. Indeed, if p(xn−1|y0:n−1) is

Gaussian, it can be shown that p(xn|y0:n) is also Gaussian provided that the model noise ηn and the observation noise εn are Gaussian, Mn−1(xn−1) is a linear function of xn−1 and Hn(xn) is a linear function of xn. In this case, (2.1) can be then rewritten as:   xn = Mn−1xn−1 + ηn−1 , (2.8)  yn = Hnxn + εn where Mn−1 of dimension and Hn, of dimensions Nx × Nx and No × Nx respectively, are known matrices that deﬁne the linear functions. Let also x0 be Gaussian and ηn and εn be Gaussian with zero means and covariances Qn and Rn, respectively. We deﬁne the conditional expectations for analysis and forecast, that minimize re- 22 spectively the a priori and the a posteriori mean square errors as [6]:

f xn = E(xn|y0:n−1), (2.9)

a xn = E(xn|y0:n), (2.10)

The conditional error analysis and forecast covariance matrices can be also deﬁned as:

f Pn = cov(xn|y0:n−1), (2.11)

a Pn = cov(xn|y0:n). (2.12)

In the following a bayesian derivation of both analysis and forecast steps of the KF will be presented.

Update (analysis) step

f f f f Assume that x0 ∼ N (x0 ,P0 ). At any time step n, xn|y0:n−1 ∼ N (xn,Pn ) i.e.

1 T −1 p(x |y ) = c exp − (x − xf ) (P f ) (x − xf ) , (2.13) n 0:n−1 1 2 n n n n n

where c1 is a constant. We also have:

1 p(y |x ) = c exp − (y − H x )T R−1(y − H x ) , (2.14) n n 2 2 n n n n n n n

where c2 is a constant. Applying the Bayes rule given by (2.7), one obtains:

1 p(x |y ) = c exp − J(x ) , (2.15) n n 3 2 n 23 with,

f T f −1 f T −1 J(xn) = (xn − xn) Pn (xn − xn) + (yn − Hnxn) Rn (yn − Hnxn). (2.16)

and c3 is a constant. J(xn) is a sum of two terms measuring the distance between

f the solution and the prior xn and between the solution and the observation weighted by their respective uncertainties as represented by the inverse of the forecast and observational error covariance matrices. Since H is linear, J(xn) is a quadratic form in xn and therefore has one global solution. Equation (2.16) can be rearranged as:

a T a −1 a J(xn) = (xn − xn) (Pn ) (xn − xn) + c4 (2.17)

where c4 is a constant and

−1 a a T −1 f −1 f a f −1 T −1 xn = Pn Hn Rn yn + (Pn ) xn and Pn = (Pn ) + Hn Rn Hn (2.18).

a a xn and Pn are respectively the mean and covariance of p(xn|y0:n). These two update moments can be written in several forms, the most common are:

a f f xn = xn + Kn(yn − Hnxn), (2.19)

where Kn is the Kalman gain given by::

f T f T −1 Kn = Pn Hn (HnPn Hn + Rn) , (2.20)

a T −1 = Pn Hn Rn . (2.21) 24 The update of the covariance can also be rewritten as:

a f f T f T −1 f Pn = Pn − Pn Hn (HnPn Hn + Rn) HnPn , (2.22)

f = (I − KnHn)Pn . (2.23)

Forecast step

f f The forecast distribution xn|y0:n−1 ∼ N (xn,Pn ). Using conditional expectation and conditional variance arguments, the mean and variance that charcterize this distribution are given, respectively, by:

f xn = E(xn|y0:n−1), (2.24) = E E(xn|xn−1)|y0:n−1 , (2.25)

= E(Mn−1xn−1|y0:n−1), (2.26)

= ME(xn−1|y0:n−1), (2.27)

a = Mn−1xn−1. (2.28) and,

f Pn = var(xn|y0:n−1), (2.29) = E var(xn|xn−1)|y0:n−1 + var E(xn|xn−1)|y0:n−1 , (2.30)

= EQn−1|y0:n−1) + var((Mn−1xn−1|y0:n−1)), (2.31)

a T = Qn−1 + Mn−1Pn−1Mn−1 . (2.32)

Summary of the KF algorithm

• Initialization:

f f x0 ∼ N (x0 ,P0 ). 25 • Analysis step:

a f f xn = xn + Kn(yn − Hnxn),

a f Pn = (I − KnHn)Pn . with,

f T f T −1 Kn = Pn Hn (HnPn Hn + Rn) .

• Forecast step:

f a xn = Mn−1xn−1,

f a T Pn = Mn−1Pn−1Mn−1 + Qn−1. The KF produces unbiased analysis and forecast of the state and their associated covariances. However, it is computationally prohibitive for large scale data assimilation problems in oceanography and atmosphere as it requires manipulating the state error

f a covariance matrices, Pn and Pn of size Nx × Nx. The KF is also designed only for linear systems, while the dynamics of these systems are nonlinear.

2.4 The (stochastic) Ensemble Kalman ﬁlter (EnKF)

The EnKF is a Monte Carlo implementation of the KF designed for large scale nonlinear systems [15]. Its key idea is to represent the distribution of the system state by a collection of realizations called ensemble. The state estimate and its error covariance matrix are then estimated as the sample mean and covariance of the ensemble. Start-

ing from a given forecast or analysis ensemble, at time tn−1, composed of Ne members

Ne e,(i) e,1 e,2 e,Ne corresponding to state vectors of dimension Nx, {xn−1}i=1 = {xn−1, xn−1, ..., xn−1 } (e can be f for forecast or a for analysis), the EnKF also operates as a succession of forecast and analysis steps.

The forecast step

Let the subscript i to denote the ith member of the considered ensemble. Starting

a,(i) Ne from an analysis ensemble {xn−1}i=1, the forecast ensemble is generated by integrating 26 every member with the model forward in time.

f,i a,i i xn = Mn−1(xn−1) + ηn i = 1, 2, ..., Ne

i where ηn ∼ N (0,Qn−1). The forecast state is then taken as the mean of the forecast ensemble: Ne f 1 X f,i xn = xn (2.33) Ne i=1

f and the associated error covariance matrix Pn as the sample covariance of the ensemble: Ne f 1 X f,i f f,i f T Pn = (xn − xn)(xn − xn) . (2.34) Ne − 1 i=1

Deﬁning the Nx × Ne ensemble perturbation matrix as

h i 0f 1 f,1 f f,2 f f,Ne f Xn = √ (xn − xn), (xn − xn), ..., (xn − xn) , (2.35) Ne − 1 the EnKF covariance matrix can be decomposed as

f 0f 0f T Pn = Xn (Xn ) . (2.36)

0f Using the perturbation matrix Xn , allows to avoid manipulating the Nx × Nx matrix

f Pn which is never explicitly computed in large scale applications.

The analysis step

In the analysis step, the EnKF applies the KF analysis step to update each member of the forecast ensemble with the incoming observation perturbed with a random noise sampled from the observational error distribution. Observations perturbations are needed to asymptotically match the expression of the analysis error covariance matrix of the KF. This is the reason why the EnKF is currently referred to as the stochastic EnKF. 27 f,i When a new observation yn is available, each member, xn of the forecast ensemble is updated as

a,i f,i i f,i xn = xn + Kn(yn − Hnxn ), (2.37)

where

i i yn = yn + γn,

i and γn is a random vector drawn from a Gaussian distribution with mean zero and

covariance Rn. The Kalman gain, Kn, is deﬁned as in (2.20) and can be decomposed as

0f 0f T 0f 0f T −1 Kn = Xn (HnXn ) ((HnXn )(HnXn ) + Rn) (2.38)

The analysis state is then taken as the sample mean:

Ne a 1 X a,i xn = xn , (2.39) Ne i=1

and the associated error covariance matrix is estimated as

Ne a 1 X a,i a a,i a T Pn = (xn − xn)(xn − xn) (2.40) Ne − 1 i=1

2.5 EnKF limitations and and auxiliary methods

2.5.1 EnKF limitations

The major issues in geophysical data assimilation problems are due to the nonlinear- ity of model dynamics, restricted ensemble sizes relatively to the dimension of the state space and poor knowledge of model errors. Unlike the linear case, when the model is nonlinear, the true forecast covariance can be exactly computed only with infinite ensemble size, even when the analysis error covariance if of low-rank. This ensemble size, however, is quite limited in realistic applications leading to approximate 28 covariances. The associated errors may degrade the filter performance and are likely to propagate to subsequent steps of the filter. It is therefore important to adequately choose the size of the ensemble: small enough to ensure reasonable computational cost especially in operational applications where timely forecasts are required, but, not too small to be statistically meaningful [18]. Undersampling also imposes two Important problems in ensemble filtering, namely, rank deficiency, issues related to long-range spurious correlations [19] and the observational error undersampling problem [10].

– Rank deﬁciency

Using small ensembles means that a significant part of the state space can not be represented by the ensemble since the null space is large meaning that there are not enough degrees of freedom to fit the data. Thus, important features may be omitted from the EnKF analysis, which might deteriorate the filter behavior and may lead to filter divergence.

– Development of long range spurious correlations

Undersampling may also cause spurious long-range correlations [20]. These correlations appear in the forecast error covariance between remote variables that are expected to be uncorrelated. The consequence is that a state variable may be in- correctly impacted by an observation that is physically remote. Empirically, at large distances, the background error covariance estimates tend to be dominated more by noise rather than the signal itself [19] .

– Observational error undersampling problem

Perturbing the observations introduces noise during the filter’s analysis. When the number of independent observations is larger than the ensemble size, the observational error covariance matrix R will be strongly undersampled and the analysis error covariances will be systematically underestimated after each assimilation cycle [10], 29 contributing to the inbreeding problem as defined by Houtekamer and Mitchell in [21]. The inbreeding problem occurs when the ensemble spread collapses which strongly limits the impact of the update step. Not accounting for model errors is also a major factor in the inbreeding problem. Neglecting the random noise in (2.8) produces unrealistic confidence in the filter forecast and can result in underestimating the background covariance matrix and the ensemble spread, which degrades the fit to observations [22].

2.5.2 Methods to mitigate EnKF limitations

Covariance inﬂation

Inflation of the ensemble spread was introduced as a straightforward way to overcome the inbreeding problem. It can be simply implemented by inflating the background forecast deviations from the mean by an inflation factor 1 + δ, as:

f,i f f,i f xn ← xn + (1 + δ)(xn − xn), (2.41)

f where xn is the forecast ensemble mean. This is equivalent to inﬂating the forecast covariance matrix by (1 + δ)2 such that:

f 2 f Pn ← (1 + δ) Pn . (2.42)

δ is typically chosen to be slightly larger than 0. The value of the inﬂation factor depends on the problem in hand, ensemble size, assumption on model and observational errors, nonlinearities ... [23] and is, very often, chosen by trial and error. Sophisticated adaptive algorithms have been also proposed for online adjustment of the value of the inﬂation factor with more or less success [24]. 30 Localization methods

While covariance inflation alleviates the issue of inbreeding, localization methods tackle the problems of rank deficiency and spurious long-range correlations [25]. To filter out spurious long-range correlations and enforce more degrees of freedom to fit the data, two standard localization methods are usually applied: the local analysis method and the covariance localization method.

– Local Analysis (LA)

The idea is to restrict the impact of an observation to nearby variables [25] under the assumption that an observation should not impact a grid point if it is located at a large distance to this observation. The formulation of LA is particularly simple, it neglects observations beyond a cut-oﬀ radius. Consequently, the analysis is performed by a sequence of local updates in disjoint sub-domains. In the most common case, where the sub-domain is a node, the analysis will be performed node by node. The LA is scheme-independent and can be applied to all schemes without exception. The algorithm of LA is presented in Appendix A.

– Covariance Localization (CL)

CL consists of cutting off long range correlations in the error covariance matrix. It modifies the covariance matrix in the Kalman gain formula by its element-wise product with some distance-based correlation matrix. The motivation behind this is to increase the rank of the covariance matrix and to cut off long range correlations between distant state vector elements. CL requires the generation of two correlation matrices C1 and C2. In the gain matrix K, the so-called cross covariance matrix

f T Pn Hn is localized by a Schur product with a matrix C1. The so-called projection

f T covariance HnP Hn is also localized by a Schur product with a second matrix C2 31 such that in the EnKF, the Kalman gain expression becomes [26] :

f T f T −1 Kn = C1 ◦ Pn Hn (C2 ◦ HnPn Hn + Rn) . (2.43)

The elements of these correlation matrices are usually calculated using a fifth-order function of Gaspari and Cohn (1999) which is defined as a correlation function with a local support, i.e. correlations decrease to zero at a finite radius as described in [26]. The algorithm of CL is presented in Appendix A. In this work, the experiments will be conducted using, exclusively, the LA algorithm with all the filters.

The hybrid EnKF

The hybrid approach has been introduced by Hamill and Snyder as a way to mitigate the background limitations of the EnKF [27]. The scheme uses a background covariance in the EnKF that is an average of a ﬂow-dependent background error covariance estimated from the EnKF ensemble and a time invariant (stationary) covariance typically used in an Optimal Interpolation (OI) or 3DVAR scheme [28]. The expression of the background covariance is then

Hybrid f Pn = (1 − α)Pn + αB, (2.44)

f where Pn is the sample forecast covariance matrix of the EnKF ensemble, B is an invariant background covariance matrix, and α is a weighting factor between 0 and 1. Obviously, when α = 0, the scheme simpliﬁes to the standard EnKF, and when α = 1 the scheme reduced to an OI scheme. The motivation behind the hybrid approach is to ﬁlter out the background structures that have large errors relative to more accurately known (climatological) background features and observations. The static background covariance, B, is often obtained from a large inventory of historical forecast errors sampled over large windows. [28]. 32 The adaptive EnKF

As the hybrid approach, the adaptive EnKF has been proposed to mitigate the background covariance limitation of the EnKF [22]. Its key idea is to adaptively enrich the EnKF ensemble by new members generated from the ensemble null space as re- constructed by the back projection of the statistics of the analysis error onto the state space The back projection is based on a stationary covariance matrix B and is implemented using an OI or a 3DVAR inversion framework. A more sophisticated version was later proposed in which the generated new ensemble directions were integrated backward in time with adjoint model to include more dynamics and observations in the back projection process following a 4DVAR approach.

Deterministic Kalman ﬁltering

Deterministic EnKFs (DEnKFs) are alternatives to the stochastic EnKF that do not use perturbed observations [11]. Thus, they eliminate a source of noise that is introduced into the problem when using the stochastic EnKF. For this reason, DEnKFs are expected to outperform the stochastic EnKF when small ensemble sizes are used with a large amount of observations [29]. In these ﬁlters, a KF update is ﬁrst applied to the mean of the forecast ensemble. Then, an analysis ensemble is sampled in order to match the analysis state estimate and the associated error covariance matrix as estimated by the KF correction step. The forecast step is then identical to that of the EnKF. DEnKs exploit the square-root low-rank nature of the EnKF sample covariance matrix to decompose the Kalman gain as:

f T f T −1 Kn = Pn Hn (HnPn Hn + Rn) , (2.45)

0f 0f T 0f 0f T −1 = Xn (HnXn ) ((HnXn )(HnXn ) + Rn) , (2.46)

0f 0f T −1 = Xn Yn Sn . (2.47) 33 with

0f 0f T Sn = Yn Yn + Rn. (2.48)

Sn is a No × No positive deﬁnite matrix and hence, is invertible. The analysis state is then computed as

a f f xn = xn + Kn(yn − Hxn). (2.49)

The analysis covariance matrix can be then decomposed as

a f Pn = (I − KnHn)Pn , (2.50)

0f 0f T −1 0f 0f T = (I − Xn Yn Sn Hn)Xn Xn , (2.51)

0f 0f T −1 0f 0f T = Xn (I − Yn Sn Yn )Xn , (2.52)

0f T 0f T = Xn VnVn Xn , (2.53)

0a 0aT = Xn Xn . (2.54)

0f T −1 0f where Vn is an Ne × Ne matrix that represents a square-root of I − Yn Sn Yn ; that is,

T 0f T −1 0f VnVn = I − Yn Sn Yn . (2.55)

The analysis ensemble members can then be computed as

a,i a 0a,i xn = xn + xn , (2.56)

0a where the perturbation vectors for are columns of Xn . The Ensemble Transform Kalman filter (ETKF) [13], the Ensemble Adjustment Kalman filter (EAKF) [14], the Ensemble Square-root Kalman filter with sequential processing of observations (EnSRF) [8] and the Singular Evolutive Interpolated Kalman filter (SEIK) [12] are different variants of ESRFs that are derived based on 34 this formulation.

2.6 Discussion

In this chapter, a Bayesian framework for data assimilation was first presented and the KF was introduced as an optimal solution for the linear Gaussian case. The stochastic EnKF was then presented as a data assimilation scheme suitable for large scale nonlinear problems. It was shown that the EnKF has many limitations mainly due to the restricted ensemble size in large scale applications, and the omission of model errors. This leads to a bad representation of the background covariance due to ensemble sampling errors and inbreeding problem. Many auxiliary techniques were proposed to mitigate these limitations. Among these techniques, inflation and localization were described and some more recent like the hybrid, the adaptive and the deterministic EnKF were also presented as auxiliary solutions. In the next chapter, we will focus on a recently introduced EnKF scheme that follows an OSA pathway to recursively compute the forecast and analysis state estimates. This formulation introduces a new update step to the state based on future observation, without violating the Bayesian filtering framework at the cost of a second forecast step of the ensemble. The new scheme is expected to improve the background approximations in an EnKF through better exploitation of the data. 35

Chapter 3

Ensemble Kalman Filtering with One-Step-Ahead Smoothing

3.1 Introduction

The classical path which involves the forecast pdf p(xn|y0:n−1) when moving from the analysis pdf p(xn−1|y0:n−1) to the analysis pdf at the next time p(xn|y0:n) is not unique [30]. One-step-ahead smoothing provides an alternative path that may improve the EnKF background through a better exploitation of the data. The algorithm has the particularity of containing two update steps rather than one. OSA smoothing was first introduced in the context of Kalman filtering; that is linear Gaussian state-space system by [31]. A stochastic EnKF variant (EnKF-OSA) was then proposed by [1], for state-parameter system, with successful application to a groundwater model. In the linear Gaussian case, the standard and the OSA-based schemes should be equivalent in the sense that they provide the same state estimates. However, in the nonlinear case, the OSA-based approach is supposed to encounter the background limitations caused by the nonlinear dynamics as well as the sampling errors and to improve the state estimates in consequence. This chapter introduces the Bayesian derivation of the OSA smoothing filtering for state estimation in both linear and nonlinear cases.

3.2 The Kalman ﬁltering with one-step-ahead smoothing (KF- OSA)

As the classical filtering scheme, the filtering with OSA smoothing addresses the problem of Bayesian estimation of the state of the system given by (2.8). However, 36 the key difference between standard and OSA smoothing Kalman filtering is that the second exploits the observations in both the state smoothing and analysis steps. Compared to the standard scheme., the OSA-based filtering scheme adds one more Kalman correction known as smoothing and one more model integration (Forecast 2). From now on, we introduce three new notations used in the OSA-based schemes. The results of the forecast step that starts from the analysis estimates will be labeled with f1. The smoothing state will be labeled with s and the results of the forecast step that starts from the smoothing estimates will be labeled with f2. We should mention here that, unlike the first forecast (f1), the second one (f2) is not a ”true” forecast since the observation yn is already used to produce the corresponding estimate.

The OSA-based ﬁltering scheme

Forecast1 a f1 {xn−1} {xn } The classical ﬁltering scheme Smoothing ) (yn

Forecast Forecast 2 a f s f2 {xn−1} {xn} {xn−1} {xn }

analysis (yn) analysis (yn)

a a {xn} {xn}

3.2.1 Generic algorithm

In the OSA-based algorithm, the analysis pdf p(xn|y0:n) is computed from p(xn−1|y0:n−1) following two steps [30] [1]:

• Smoothing step: The one-step-ahead smoothing pdf, p(xn−1|y0:n), is ﬁrst computed as:

p(xn−1|y0:n) ∝ p(yn|xn−1, y0:n−1)p(xn−1|y0:n−1) (3.1) 37 with

The smoothing step is a measurement-update step where p(xn−1|y0:n−1) is up-

dated using the new observation yn, yielding the OSA smoothing pdf p(xn−1|y0:n).

• Analysis step: The analysis pdf p(xn|y0:n) is computed from the smoothing pdf

at time step n − 1. Using the a posteriori transition pdf p(xn|xn−1, y0:n), one obtains: Z p(xn|y0:n) = p(xn|xn−1, y0:n)p(xn−1|y0:n)dxn−1, (3.3)

with

p(xn|xn−1, y0:n) ∝ p(yn|xn)p(xn|xn−1). (3.4)

3.2.2 KF-OSA equations

Consider again the linear discrete-time dynamical system (2.8):

  xn = Mn−1xn−1 + ηn−1 , (3.5)  yn = Hnxn + εn

In order to simplify the notations, let’s deﬁne the following quantities:

f1 f1 xn = E(xn|y0:n−1) Pn = (xn|y0:n−1)

s s xn = E(xn−1|y0:n) Pn−1 = (xn−1|y0:n)

a a xn = E(xn|y0:n) Pn = (xn|y0:n)

6 6 In the linear Gaussian case, the expressions of the OSA-KF as presented in [30] are 38 given by:

– Forecast (1) step: The analysis estimate at time tn−1 is integrated forward in

time using the model to obtain the forecast (1) estimate at time tn.

f1 a xn = Mn−1xn−1 (3.6)

f1 a T Pn = Mn−1Pn−1Mn−1 + Qn−1 (3.7)

– Smoothing step: The forecast (1) estimate and the observation at time tn are

used to smooth the analysis estimate at the previous time tn−1.

−1 s a a T T f1 T f1 xn−1 = xn−1 + Pn−1Mn−1 Hn (HnPn Hn + Rn) (yn − Hnxn ) (3.8)

−1 s a a T T f1 T a Pn−1 = Pn−1 − Pn−1Mn−1 Hn (HnPn Hn + Rn) HnMn−1Pn−1 (3.9)

– Forecast (2) step: The smoothed estimate at time tn−1 is integrated forward

in time using the model to obtain the forecast (2) estimate at time tn.

f2 s xn = Mn−1xn−1 (3.10)

– Analysis step: The observation at time tn is used to update the forecast (2) estimate.

−1 a f2 T f1 T f1 xn = xn + Qn−1Hn (HnPn Hn + Rn) (yn − Hnxn ) (3.11)

−1 f2 T T f2 = xn + Qn−1Hn (HnQn−1Hn + Rn) (yn − Hnxn ) (3.12) | {z } Kgn a s T Pn = Mfn−1Pn−1Mfn−1 + Qen−1 (3.13) 39

Where Mfn−1 and Qen−1 are given by:

Mfn−1 = (I − KenHn)Mn−1 (3.14)

Qen−1 = (I − KenHn)Qn−1 (3.15)

3.3 Ensemble formulation (EnKF-OSA)

When the model M(.) is nonlinear (In this case, it is represented as M(.)), it is generally not possible to ﬁnd an analytical solution for the problem described in section 3.2.1. An OSA smoothing based ensemble Kalman formulation was then proposed in [1] for the more general state-parameter estimation problem. The goal was to derive a Bayesian consistent dual-like state parameter Kalman ﬁlter. The ensemble formulation is derived using the following random sampling properties:

Property 1 (Hierarchical sampling) [32]. Assuming that one can sample from p(x1)

∗ and p(x2|x1), then a sample, x2, from p(x2) can be drawn as follows:

∗ 1. x1 ∼ p(x1);

∗ ∗ 2. x2 ∼ p(x2|x1).

Property 2 (Conditional sampling) [33]. Consider a Gaussian pdf, p(x, y), with Pxy and Py denoting the cross-covariance of x and y and the covariance of y, respectively. Then a sample, x∗, from p(x|y), can be drawn as follows:

1.( x,e ye) ∼ p(x, y);

∗ −1 2. x = xe + PxyPy [y − ye]. 40 3.3.1 Smoothing step

f1,(i) In this step, the forecast members, xn , resulting from the forecast (1) step are used with the observation yn to update the analysis members at the previous time step,

a,(i) s,(i) xn−1 generating the smoothing ensemble, xn−1. Starting from an analysis ensemble, a,(i) Ne {xn−1}i=1 at time tn − 1 , Property 1 can be used in equation (3.2) to sample the N f1,(i) e observation forecast ensemble, {yn }i=1, as:

f1,(i) a,(i) (i) xn = Mn−1(xn−1) + ηn−1, (3.16)

f1,(i) f1,(i) (i) yn = Hnxn + εn , (3.17)

(i) (i) with ηn−1 ∼ N (0,Qn−1) and εn ∼ N (0,Rn). Then, to obtain the smoothing ensem- s,(i) Ne ble, {xn−1}i=1, Property 2 can be used in Equation (3.1) to sample:

s,(i) a,(i) −1 f1,(i) f xn−1 = xn−1 + P a 1 P f1 yn − yn , (3.18) xn−1,yn yn

where the cross-covariance term is evaluated from the analysis and forecast (1) ensembles according to:

0 0 T −1 a f1 P a f1 = (Ne − 1) Xn−1Yn (3.19) xn−1,yn

In other words, the model is ﬁrst used to compute the state forecast ensemble, N N f1,(i) e a,(i) e {xn }i=1, from the previous analysis ensemble, {xn−1}i=1, as in (3.16). The ob- N f1,(i) e servation forecast ensemble, {yn }i=1, can be then obtained using (3.17). This N f1,(i) e ensemble, {yn }i=1, serves with the current observation, yn, to correct the analysis a,(i) Ne ensemble, {xn−1}i=1, following the EnKF principle as highlighted in (3.18). 41 3.3.2 Analysis step

a,(i) Ne Using equation (3.3), the most natural way to obtain the analysis members {xn }i=1 would be to apply Property 1 in equation (3.3). That is, once the smoothing mem-

s,(i) Ne bers {xn−1}i=1 are generated, one may sample from the a posteriori transition pdf, p(xn|xn−1, yn), using equation (3.4). Indeed, it can be shown that equation (3.4) leads to a Gaussian pdf as given by equation (3.20):

p(xn|xn−1, yn) = Nxn Mn−1(xn−1) + Ken (yn − HnMn−1(xn−1)) , Qen−1 , (3.20)

T T −1 with Ken = Qn−1Hn HnQn−1Hn + Rn and Qen−1 = (I − KenHn)Qn−1.

Deﬁning,

f2,(i) s,(i) (i) xn = Mn−1(xn−1) + ηn−1, (3.21)

f2(i) f2,(i) (i) yn = Hnxn + εn , (3.22)

(i) (i) with ηn−1 ∼ N (0,Qn−1) and εn ∼ N (0,Rn), one can show that samples from p(xn|y0:n) can be drawn as:

a,(i) f2,(i) f2(i) xn = xn + Ken(yn − yn ) (3.23)

However, these members can not be generated under this form for two main rea- sons. First, when we deal with high-dimensional problems, the computational cost of Ken and Qen−1 (which may be an oﬀ-diagonal matrix even if Qn−1 is diagonal) becomes very expensive. Another serious limitation lies in the fact that in realistic ocean-atmosphere applications, the model error covariance matrix Qn−1 is usually very poorly known. To avoid the manipulation of the model error covariance matrix in the analysis equa- 42

tions was presented in [1] and consists in directly sampling from p(xn|xn−1, yn) rather than explicitly computing this pdf through its expression given by (3.20). More pre-

cisely, to sample from p(xn|xn−1, yn), one may apply directly Properties 1 and 2 to (3.4).

(i) Ne To do this, let {xn (xn−1)}i=1 denotes an ensemble of samples drawn from p(xn|xn−1, yn). (i) (i) The notation xn (xn−1) refers to a function xn of xn−1. Similar notations hold for

f2,(i) f2(i) xn (.) and yn (.) in (3.24) and (3.25), respectively. Using Properties 1 and 2, an explicit form of such samples can be obtained as [1]:

f2,(i) (i) (i) xn (xn−1) = Mn−1(xn−1) + ηn−1; ηn−1 ∼ N (0,Qn−1), (3.24)

f2(i) f2,(i) (i) (i) yn (xn−1) = Hnxn (xn−1) + εn ; εn ∼ N (0,Rn), (3.25)

(i) f2,(i) −1 f2(i) f f xn (xn−1) = xn (xn−1) + P 2 2 P f2 yn − yn (xn−1) , (3.26) xn ,yn yn

where the covariances, P f2 f2 and P f2 , can be approximated from the ensembles xn ,yn yn N N f2,(i) e f2(i) e {xn (xn−1)}i=1 and {yn (xn−1)}i=1. The next step would be to apply Property 1 to equation (3.3) as discussed previously in order to compute an analysis ensem-

a,(i) Ne s,(i) Ne ble, {xn }i=1, from the smoothing ensemble, {xn−1}i=1. This is done by setting s,(i) a,(i) (i) s,(i) xn−1 = xn−1 in equations (3.24)-(3.26), in order to obtain xn = xn (xn−1).

3.4 Summary of the EnKF-OSA algorithm

a,(i) Ne Starting at time step n − 1 from an analysis ensemble, {xn−1}i=1, the updated ensemble at the following time step n is obtained with the following two steps:

N f1,(i) e • Smoothing step: The forecast ensemble members, {xn }i=1, are ﬁrst generated N f1,(i) e using (3.16). Then, the observation forecast ensemble, {yn }i=1, is computed using (3.17). Both observation forecast ensemble and available observation at

time step n, yn, are then used to compute the one-step-ahead smoothing en- 43 s,(i) Ne semble, {xn−1}i=1, through (3.18).

a,(i) Ne • Update step: The analysis ensemble {xn }i=1 is obtained from the smoothing s,(i) Ne ensemble {xn−1}i=1 using:

f2,(i) s,(i) (i) (i) xn = Mn−1(xn−1) + ηn−1; ηn−1 ∼ N (0,Qn−1) (3.27)

f2(i) f2,(i) (i) (i) yn = Hnxn + εn ; εn ∼ N (0,Rn) (3.28)

a,(i) f2,(i) −1 f2(i) f f xn = xn + P 2 2 P f2 yn − yn . (3.29) xn ,yn yn

3.5 Discussion

In this chapter, the concept of the one-step-ahead smoothing based ﬁltering algorithms was recalled. The linear Gaussian case, for which optimal KF algorithm was recalled, was ﬁrst presented. Then, the nonlinear case, for which the stochastic ensemble formulation (EnKF-OSA), was outlined. As discussed previously, deterministic EnKF have some advantages over the stochastic EnKF mainly when the ensemble size is limited compared to the number of assimilated data, which represents a common case in realistic ocean-atmosphere problems. One may expect this fact to remain true with the OSA smoothing based version which explains the need to develop a deterministic OSA-smoothing based EnKF in order to avoid the observational error undersampling problem in large scale applications. In the following chapter, problems related to the deterministic formulation will be explained and two solutions will be proposed. 44

Chapter 4

Deterministic Ensemble Kalman Filtering with One-Step-Ahead Smoothing

4.1 Introduction

Deterministic EnKFs were introduced to avoid perturbing the observations. They are based on the KF equations and produce an analysis ensemble having a mean and a covariance that exactly match those defined by the KF equations. As discussed in Chapter 3, the OSA-based algorithm applies two updates with the same observation: one for smoothing and one for analysis. In contrast with the smoothing step, the KF-OSA equations for the analysis step are functions of the model error covariance matrix Q, and therefore require the knowledge of this matrix to compute the analysis state estimate and its error covariance. Q is, however, poorly known in realistic ocean-atmosphere applications. We introduce, in this chapter, two new deterministic variants of the EnKF-OSA in which we avoid the manipulation of Q. The two proposed algorithms have the same smoothing step and differ only at the level of the analysis step. The first algorithm avoids Q by parametrizing it as a fraction of the forecast error covariance. The second algorithm is inspired from the stochastic EnKF-OSA equations; it rewrites the same analysis equations under a form where Q is hidden. We present the two algorithms in the context of the Singular Evolutive Interpolated Kalman (SEIK) filter. The algorithms can be, however, adapted to any other deterministic EnKF. 45 4.2 The Singular Evolutive Interpolated Kalman filter (SEIK)

In this part, we will recall the algorithm of the SEIK filter as it will be used later in the elaboration of the algorithms and in the experiments. Originally, the SEIK filter was introduced in [34] and was discussed in more details in [35]. Like all square root filters, the SEIK filter respects the original separation of the analysis step into mean state update (analysis) and ensemble transformation (resampling).

Mean state analysis

f In the SEIK ﬁlter, the forecast covariance matrix Pn is given by:

f f T −1 f T Pn = Ln[(Ne − 1)T T ] Ln (4.1) where,

f f Ln = Xn T (4.2) and, 1 −1 G = (T T T ) (4.3) Ne − 1

T is an Ne ×(Ne −1) matrix with full rank and zero column sums. In previous studies [11], [36], the matrix T has always been deﬁned as:

  I(Ne−1)×(Ne−1)   - 1 1   Ne Ne×(Ne−1) 0 1×(Ne−1) (4.4) 46 f With this choice of the matrix T, Ln will be simply represented as:

f f,1 f,2 f,Ne−1 f,Ne Ln = [xn , xn , ..., xn , xn ]T (4.5)

f,1 f f,2 f f,Ne−1 f = [xn − xn, xn − xn, ..., xn − xn] (4.6)

This means that the matrix T implicitly subtracts the ensemble mean from the fore-

f cast ensemble members and also removes the last column. Therefore, Ln will be an

Nx × (Ne − 1) matrix that simply holds the ﬁrst Ne − 1 forecast ensemble perturbations.

f Using this form of Pn , the Kalman gain, originally given by equation (2.20), can be written as:

f a f T −1 Kn = LnUn (HnLn) Rn (4.7) where,

a −1 −1 f T −1 f (Un ) = G + (HnLn) Rn HnLn (4.8)

The mean forecast state estimate is then updated leading to a mean analysis estimate

f which is a combination of the columns of the matrix Ln by:

a f f xn = xn + Lnan (4.9) with,

a f T −1 f an = Un (HnLn) Rn (yn − Hnxn) (4.10) 47 Although it does not need to be explicitly computed, the expression of the analysis covariance matrix is given in factorized form by:

a f Pn = (I − KnHn)Pn (4.11)

f T f T −1 f = (I − Pn Hn (HnPn Hn + Rn) Hn)Pn (4.12) −1 f f T T f f T T f f T = (I − LnGLn Hn (HnLnGLn Hn + Rn) Hn)LnGLn (4.13) −1 f h f T f f T f i f T = Ln G − G(HnLn) (HnLnG(HnLn) + Rn) HnLnG Ln (4.14) −1 f h −1 f T −1 f i f T = Ln G + (HnLn) Rn (HnLn) Ln (4.15)

f a f T = LnUn Ln (4.16)

Equation (4.51) is derived from (4.14) using the matrix inversion lemma.

Resampling step

a After updating the mean analysis state, xn, using the available observation yn, the resampling of the analysis ensemble members is performed. The transformation is done such that the generated analysis ensemble has a mean and a covariance that

a a exactly match xn and Pn given by (4.9) and (4.16), respectively. In the SEIK ﬁlter, the resampling step is performed according to:

a,i a p f a −1T T xn = xn + Ne − 1Ln(Cn) Ωn(:,i) (4.17)

where (:, i) stands for the ith column of the considered matrix. In (4.17), the matrix

a a −1 a Cn is a square-root of the matrix (Un ) . Usually, Cn is obtained through a Cholesky

a −1 a aT a decomposition of the matrix (Un ) to CnCn to enable to write Pn as:

a f a −1 T T a −1 f T Pn = Ln((Cn) ) Ωn Ωn(Cn) (Ln) (4.18) 48 Other forms of square-roots rather than the Cholesky decomposition may be used here. The matrix Ωn is an Ne × Ne − 1 matrix with orthonormal columns and zero column sums. The procedure to generate the random matrix Ωn is described in [35].

4.3 SEIK ﬁlter with One-Step-Ahead smoothing (SEIK-OSA)

The OSA smoothing based SEIk ﬁlter is a deterministic square-root formulation of the KF-OSA based on the SEIK equations.

4.3.1 Smoothing step

We start from the KF-OSA equations for the smoothing step.

−1 s a a T T f1 T f1 xn−1 = xn−1 + Pn−1Mn−1 Hn (HnPn Hn + Rn) (yn − Hnxn ) (4.19)

−1 s a a T T f1 T a Pn−1 = Pn−1 − Pn−1Mn−1 Hn (HnPn Hn + Rn) HnMn−1Pn−1 (4.20)

We deﬁne the OSA smoothing Kalman gain as:

−1 s a T T f1 T Kn−1 = Pn−1Mn−1 Hn (HnPn Hn + Rn) (4.21)

To derive an ensemble formulation of the smoothing state, we start at time tn−1, from N N a,(i) e f1,(i) e an analysis ensemble, {xn−1}i=1. The forecast members, {xn }i=1 are obtained, as a,(i) Ne in the standard EnKF, by integrating {xn−1}i=1 forward with the nonlinear model

Mn−1.

f1,(i) a,(i) (i) xn = Mn−1xn−1 + ηn−1 (4.22) 49 The mean analysis state is computed as:

Ne a 1 X a,(i) xn−1 = xn−1, (4.23) Ne i=1 and the mean forecast state as:

1 Ne f1 X f1,(i) xn = xn . (4.24) Ne i=1

Let also,

a a a T Pn−1 = Ln−1GLn−1 (4.25)

T f1 f1 f1 Pn = Ln GLn (4.26) with

a a Ln−1 = Xn−1T, (4.27)

f1 f1 Ln = Xn T, (4.28) and 1 −1 G = (T T T ) . (4.29) Ne − 1

s T is deﬁned in the same way as in Equation (4.4). Kn−1 can be then expressed as:

s −1 Kn−1 = P a f P f , (4.30) xn−1,yn yn T h T i−1 a f1 f1 f1 = Ln−1G(HnLn ) HnLn G(HnLn ) + Rn , (4.31)

h T i−1 T a −1 f1 −1 f1 f1 −1 = Ln−1 G + (HnLn ) Rn HnLn (HnLn ) Rn . (4.32)

| {zs } Un−1 50

The analysis mean state is then smoothed using the observation yn according to:

T s a a s f1 −1 f1 xn−1 = xn−1 + Ln−1Un−1(HnLn ) Rn (yn − Hnxn ). (4.33)

Here, we will deﬁne an approximation to derive the expression of the smoothing covari-

h a i ance. Without abusing the notation, we deﬁne Mn−1 Ln−1 such that the operator

a h a i Mn−1 operates on each column of Ln−1. Considering this notation, Mn−1 Ln−1 can

f1 be approximated, as in [37], by Ln :

h a i h a,1 a,Ne i Mn−1 Ln−1 ≡ [Mn−1(xn−1), ..., Mn−1(xn−1 )]T h i f1,1 f1,Ne ≈ [xn , ..., xn ]T

f1 ≈ Ln (4.34)

The expression of the smoothing error covariance matrix is given by equation (4.20) in the linear Gaussian case. In the ensemble formulation and using (4.25), (4.26) and

s (4.34), one can decompose Pn−1 as:

h T i−1 s a a T a a T T f1 f1 Pn−1 = Ln−1GLn−1 − Ln−1G(Mn−1Ln−1) Hn HnLn G(HnLn ) + Rn

a a T × HnMn−1Ln−1GLn−1 ,

T h T i−1 a f1 T f1 f1 f1 a T = Ln−1 G − GLn Hn HnLn G(HnLn ) + Rn HnLn G Ln−1 ,

h T i−1 a −1 f1 −1 f1 a T = Ln−1 G + (HnLn ) Rn HnLn Ln−1 ,

a s a T = Ln−1Un−1Ln−1 .

To summarize, the smoothing ensemble mean is computed using

T s a a s f1 −1 f1 xn−1 = xn−1 + Ln−1Un−1(HnLn ) Rn (yn − Hnxn ), (4.35) 51 and the corresponding smoothing error covariance matrix is deﬁned as:

s a s a T Pn−1 = Ln−1Un−1Ln−1 . (4.36)

The smoothing ensemble members are then generated exactly as in the SEIK.

s,i s p a s −1T T xn = xn + Ne − 1Ln−1(Cn−1) Ωn(:,i), (4.37)

s s −1 where Cn−1 is a square-root of the matrix (Un−1) such that:

s −1 s s T (Un−1) = Cn−1Cn−1 (4.38)

4.3.2 Analysis step

We start from the analysis step of the KF-OSA:

−1 a f2 T f1 T f1 xn = xn + Qn−1Hn (HnPn Hn + Rn) (yn − Hnxn ) (4.39)

−1 f2 T T f2 = xn + Qn−1Hn (HnQn−1Hn + Rn) (yn − Hnxn ) (4.40) | {z } Kgn a s T Pn = Mfn−1Pn−1Mfn−1 + Qen−1 (4.41)

Where Mfn−1 and Qen−1 are given by:

Mfn−1 = (I − KenHn)Mn−1 (4.42)

Qen−1 = (I − KenHn)Qn−1 (4.43)

As mentioned previously, the main diﬃculty is applying the update in the direction of the poorly known Q and the calculations of Q. We will present two solutions to derive practical analysis equations. A ﬁrst straightforward way to overcome this problem, following [12], consists in parametrizing Q as a fraction of the forecast error 52 covariance matrix estimated from the forecast members. Another way would be to project model errors on Lf , as in the Singular Evolutive Extended Kalman (SEEK)

f q f T ﬁlter, by decomposing Q as Qn−1 = LnUn(Ln) . The second solution is inspired from the stochastic formulation of the EnKF-OSA where the Kalman gain, originally containing Q, is reformulated in a way that makes the Q implicitly accounted for in the new expression.

Approximation of the model error covariance matrix (SEIK-

OSAQˆ)

One solution that avoids the use of the model error covariance matrix from the analysis equations would be to approximate this matrix using known quantities. In this ﬁrst

part, we chose to express Qn−1 as a fraction of the forecast error covariance at time

f1 step n. That is we choose to approximate Qn−1 by αPn where 0 ≤ α ≤ 1. One way

f1 to justify this approximation is to recall the deﬁnitions of Qn−1 and Pn :

Qn−1 = cov(xn|xn−1, y0:n−1) (4.44)

f1 Pn = cov(xn|y0:n−1) (4.45)

From these deﬁnitions, one may remark that Qn−1 is supposed to be associated to

f1 a more accurate estimate of xn than Pn since Qn−1 is based on more information

f1 f1 (xn−1) than Pn ; one may then expect Qn−1 to be smaller than Pn , whence α ≤ 1. s,(i) Ne After generating the smoothing ensemble {xn−1}i=1 from the smoothing step, these N f2,(i) e members are integrated forward in time by the model to generate {xn }i=1 such that:

f2,(i) s,(i) (i) (i) xn = Mn−1(xn−1) + ηn−1; ηn−1 ∼ N (0,Qn−1) (4.46) 53 Equation (4.39), is then applied to update the resulting mean state according to:

−1 a f2 T f1 T f1 xn = xn + Qn−1Hn (HnPn Hn + Rn) (yn − Hnxn ); (4.47)

f2 where xn is deﬁned as:

1 Ne f2 X s,(i) (i) xn = (Mn−1(xn−1) + ηn−1) (4.48) Ne i=1

f1 f1 and xn is given by (4.24). Replacing Qn−1 by αPn in equation (4.47), the mean state analysis equation becomes:

−1 a f2 f1 T f1 T f1 xn = xn + αPn Hn (HnPn Hn + Rn) (yn − Hnxn ) (4.49) | {z } a Kgn

a The expression of Kfn is similar to the expression of the Kalman gain in the standard scheme, but is multiplied by the tunable factor α. The analysis error covariance matrix given by (4.41) in the linear case, contains also Q in its expression. Under this form, it is not possible to ﬁnd a square-root form of this matrix unless Qn−1 is known and already decomposed in square-root. Under the

f1 approximation Qn−1 = αPn , equation (4.42) becomes:

Mfn−1 = (I − KenHn)Mn−1

T T −1 = (I − Qn−1Hn (HQn−1Hn + Rn) Hn)Mn−1

−1 f1 T f1 T = (I − αPn Hn (HnαPn Hn + Rn) Hn)Mn−1 −1 f1 T f1 T = (I − Pn Hn (HPn Hn + Rfn) Hn)Mn−1 (4.50)

Rn where Rfn = . α In the context of a nonlinear model and ensemble formulation, and using (4.26), (4.34) 54 and (4.36), the left hand side term of equation (4.41) can be approximated by:

T −1 T −1 f1 T f1 T f1 s f1 f1 T f1 T T1 = (I − Pn Hn (HPn Hn + Rfn) Hn) Ln Un−1Ln (I − Pn Hn (HPn Hn + Rfn) Hn) ,

T −1 f1 f1 f1 T f1 s = Ln I − G(HnLn ) (HPn Hn + Rfn) HnLn Un−1

T T −1 T f1 f1 T f1 f1 × I − G(HnLn ) (HPn Hn + Rfn) HnLn Ln ,

T f1 1 f1 = Ln SnLn .

The right-hand-side term of equation (4.41) becomes:

T2 = Qen−1,

T T −1 = (I − Qn−1Hn (HQn−1Hn + Rn) Hn)Qn−1,

−1 f1 T f1 T f1 = (I − αPn Hn (HnαPn Hn + Rn) Hn)αPn ,

−1 T f1 T f1 T f1 f1 = α(I − Pn Hn (HnPn Hn + Rfn) Hn)Ln GLn ,

T T −1 T f1 f1 f1 f1 f1 f1 = αLn [G − G(HnLn ) (HnLn G(HnLn ) + Rfn) HnLn G]Ln ,

T −1 −1 T f1 −1 f1 f1 f1 = αLn [G − (HnLn ) Rfn (HnLn )] Ln ,

T f1 2 f1 = αLn SnLn .

The analysis covariance matrix is then:

T T a f1 1 f1 f1 2 f1 Pn = Ln SnLn + αLn CnLn , (4.51)

h i T f1 1 2 f1 = Ln Sn + αSn Ln , (4.52)

T f1 a f1 = Ln Ufn Ln . (4.53)

a a aT Performing the Cholesky decomposition of the matrix Ufn to CfnCgn ,

T a f1 a T aT f1 Pn = Ln CfnΩn ΩnCgn Ln (4.54) 55 And then, the analysis members could be resampled according to:

a,i a p f1 a T xn = xn + Ne − 1Ln CfnΩn(:,i) (4.55)

Formulation inspired from the EnKF-OSA algorithm (SEIK- OSA)

Here, we present a deterministic formulation of the analysis step inspired from the equations of the stochastic EnKF-OSA algorithm. The stochastic EnKF-OSA algorithm provides an alternative to the analysis equation (3.23) that contains Q by another equation (3.29) where Q is implicitly accounted for. These two equations will be recalled here:

−1 a,(i) f2,(i) T T f2(i) xn = xn + Qn−1Hn (HnQn−1Hn + Rn) (yn − yn ), (4.56) | {z } Ken a,(i) f2,(i) −1 f2(i) f f xn = xn + P 2 2 P f2 yn − yn , (4.57) xn ,yn yn

T T −1 We, thus, remark that the Kalman gain Ken which is originally given by Qn−1Hn (HnQn−1Hn + Rn)

−1 f f can be expressed diﬀerently by P 2 2 P f2 when the ensemble size is large enough. xn ,yn yn This fact was deduced from the stochastic implementation and in the following, we will ﬁrst show that this new expression holds also in the linear Gaussian case using the estimates (members means) instead of the members in (4.57). We will derive the SEIK-OSA equations based on (4.56) and (4.57).

Linear Gaussian case

In the KF-OSA equations, the analysis equation can be expressed by two equations:

−1 a f2 T f1 T f1 xn = xn + Qn−1Hn (HnPn Hn + Rn) (yn − Hnxn ) (4.58)

−1 f2 T T f2 = xn + Qn−1Hn (HnQn−1Hn + Rn) (yn − Hnxn ) (4.59) 56

f2 s f2 f2 where xn = Mn−1xn−1 and yn = Hnxn . In this section, we will use equation (4.59). Since we are dealing with the linear

f2 Gaussian case, the expression of xn can be expressed as:

f2 s xn = Mn−1xn−1,

−1 a a T T f1 T f1 = Mn−1 xn−1 + Pn−1Mn−1 Hn (HnPn Hn + Rn) (yn − Hnxn ) ,

−1 a a T T f1 T f1 = Mn−1xn−1 + Mn−1Pn−1Mn−1 Hn (HnPn Hn + Rn) (yn − Hnxn ), | {z } f1 (Pn −Qn−1) −1 f1 f1 T f1 T f1 = xn + Pn − Qn−1 Hn (HnPn Hn + Rn) (yn − Hnxn ),

f1 f12 f1 = xn + Kn (yn − Hnxn ). (4.60)

−1 f12 f1 T f1 T with Kn = Pn − Qn−1 Hn (HnPn Hn + Rn) .

– Alternative expression to the Kalman gain

T T −1 −1 f f The goal of this part is to show the equality Qn−1Hn (HnQn−1Hn + Rn) = P 2 2 P f2 xn ,yn yn

f2 in the linear Gaussian case. Equation (4.60) shows that xn can be interpreted as an

f1 f12 analysis of xn using the Kalman gain Kn .

f2 y2 f2 f2 Let en and en be the errors associated respectively with the estimates xn and yn such that:

f2 f2 en = xn − xn (4.61) and,

y2 f2 en = yn − yn (4.62)

f2 = Hxn + εn − Hxn (4.63)

f2 = Hnen + εn (4.64) 57

f2 Replacing xn by its expression given by (4.60) in (4.61), we get:

f2 f2 en = xn − xn (4.65) f1 f12 f1 = xn − xn + Kn (yn − Hnxn ) (4.66) f1 f12 f1 = xn − xn + Kn (Hxn + εn − Hnxn ) (4.67)

f12 f1 f12 = (I − Kn Hn)(xn − xn ) − Kn εn (4.68)

Consequently,

y2 f2 en = Hnen + εn (4.69) f12 f1 f12 = Hn (I − Kn Hn)(xn − xn ) − Kn εn + εn (4.70)

f12 f1 f12 = Hn(I − Kn Hn)(xn − xn ) + (I − HnKn )εn (4.71)

The cross-covariance term P f2 f2 is equal to: xn ,yn

f2 y2 T P f2 f2 = (e e ) xn ,yn E n n f12 f1 f12 = E (I − Kn Hn)(xn − xn ) − Kn εn T f12 f1 f12 × Hn(I − Kn Hn)(xn − xn ) + (I − HnKn )εn

Using,

T f1 f1 f1 E((xn − xn )(xn − xn ) ) = Pn

f1 T E((xn − xn )εn ) = 0 58 T T f12 f1 f12 T f12 f12 P f2 f2 = (I − K Hn)P (I − K Hn) H − K Rn(I − K Hn) xn ,yn n n n n n n T f1 T f12 f1 T f12 = Pn Hn − Kn (HPn Hn + Rn) I − HKn T f1 T f1 T f12 = Pn Hn − (Pn − Qn−1)Hn I − HKn T T f12 = Qn−1Hn I − HKn (4.72)

The covariance term P f2 is deﬁned as: yn

y2 y2 T P f2 = (e e ) yn E n n

f2 y2 T = E((Hen + εn)en )

f2 y2 T y2 T = HnE(en en ) + E(εnen ) h iT f12 f1 f12 = HnP f2 f2 + εn Hn(I − K Hn)(xn − x ) + (I − HnK )εn xn ,yn E n n n T T T f12 f12 = HnQn−1Hn (I − HKn ) + Rn I − HKn

T T f12 = (HnQn−1Hn + Rn)(I − HKn ) (4.73)

Finally, using (4.72) and (4.73), we ﬁnd that

−1 f f P 2 2 P f2 = Kfn (4.74) xn ,yn yn T T −1 = Qn−1Hn (HQn−1Hn + Rn) (4.75)

Equation (4.75) gives an expression of the Kalman gain used in the analysis step that

does not require the matrix Qn−1 to be known. The analysis can then be written as:

a f2 f2 xn = xn + Kfn(yn − Hnxn ) (4.76)

with,

−1 f f Kfn = P 2 2 P f2 (4.77) xn ,yn yn 59 – Approximate expressions of the Kalman gain and the analysis error covariance

In this part, we will deﬁne an approximation to obtain expressions of the Kalman

f2 gain and the analysis covariance similar to those of the standard SEIK. Let Pn be the

f2 error covariance matrix associated to the estimate xn . The Kalman gain expression given by equation (4.77) can be further developed using (4.61) and (4.64). Indeed,

f2 y2 T P f2 f2 = (e e ) xn ,yn E n n T f2 f2 = E en (Hen + εn)

T f2 f2 T f2 T = E(en en )Hn + E(en εn )

f2 T = P H + P f2 n n xn ,εn

y2 y2 T P f2 = (e e ) yn E n n T f2 f2 = E (Hen + εn)(Hen + εn)

f2 T T = HnP H + Rn + HnP f2 + P f2 H n n xn ,εn εn,xn n

Finally,

−1 f2 T f2 T T Kn = P H + P f2 HnP H + Rn + HnP f2 + P f2 H (4.78) f n n xn ,εn n n xn ,εn εn,xn n

f2 where P f2 is the cross-covariance between x and εn. εn,xn n In the following part, we will derive an expression of the analysis error covariance. 60 To do so, let:

a a en = xn − xn (4.79) f2 f2 = xn − xn + Kfn(yn − Hnxn ) (4.80)

f2 = (I − KfnHn)(x − xn ) − Kfnεn (4.81) be the analysis error at time step n. The analysis error covariance matrix is given by:

a a a T Pn = E(enen ) T f2 f2 = E (I − KfnHn)(x − xn ) − Kfnεn (I − KfnHn)(x − xn ) − Kfnεn

T T f2 = (I − KfnHn)Pn (I − KfnHn) + KfnRnKfn T T T + (I − KnHn)P f2 Kn − Kn P f2 (I − KnHn) f xn ,εn f f εn,xn f T T f2 f2 T f2 = (I − KfnHn)Pn + Kfn(HnPn Hn + Rn)Kfn − Pn (KfnHn) (4.82) T T − KnP f2 (I − KnHn) − (I − KnHn)P f2 (I − KnHn)Kn f xn ,εn f f εn,xn f f

f2 Equation (4.82) gives the expression of the analysis covariance in terms of Pn and

P f2 . In order to simplify this equation and have a common form, we will suppose εn,xn here that the term P f2 is negligible. Under this assumption, the approximate xn ,εn ˆ expression Kfn of Kfn becomes:

−1 f2 T f2 T ˆ Kfn ≈ Pn Hn (HnPn Hn + Rn) = Kfn (4.83)

T T a ˆ f2 ˆ f2 T ˆ f2 ˆ Pn ≈ (I − KfnHn)Pn + Kfn(HnPn Hn + Rn)Kfn − Pn (KfnHn) (4.84)

ˆa = Pn (4.85) 61 ˆ Replacing Kfn by its expression given by (4.83) in (4.84), we obtain:

T −1 ˆa ˆ f2 f2 T f2 T f2 T ˆ Pn = (I − KfnHn)Pn + Pn Hn (HnPn Hn + Rn) (HnPn Hn + Rn)Kfn T f2 ˆ − Pn (KfnHn) T T ˆ f2 f2 T ˆ f2 T ˆ = (I − KfnHn)Pn + Pn Hn Kfn − Pn Hn Kfn

ˆ f2 = (I − KfnHn)Pn (4.86)

From (4.86), we see that under the assumption that the cross covariance term between

f2 xn and the observational error εn is not signiﬁcant, we can derive an equation of the

a analysis covariance matrix Pn , having the same form as in the standard analysis al-

a f gorithm where the analysis covariance expression is given by: Pn = (I − KnHn)Pn . ˆ Under the same approximation, Kfn , the approximate expression of Kfn is similar in its form to the Kalman gain used during the standard analysis scheme. The new approximate KF-OSA equations related to the analysis step as derived here can be summarized as follows:

Approximate KF-OSA equations for the analysis step:

a f2 ˆ f2 xn ≈ xn + Kfn(yn − Hnxn ) (4.87)

a ˆ f2 Pn ≈ (I − KfnHn)Pn (4.88)

Ensemble formulation

In this part, a deterministc ensemble formulation (based on the approximate KF-

s,(i) Ne OSA equations for the analysis step) will be derived. Let {xn−1}i=1 be the smoothing ensemble at time step n − 1. These members are integrated in time by the model to 62 N f2,(i) e generate {xn }i=1 such that:

f2,(i) s,(i) (i) (i) xn = Mn−1(xn−1) + ηn−1; ηn−1 ∼ N (0,Qn−1) (4.89)

Based on (4.87), the mean state analysis equation can be written as:

a f2 ˆ f2 xn = xn + Kfn(yn − Hnxn ) (4.90)

f2 Where xn is estimated as:

1 Ne f2 X s,(i) (i) xn = (Mn−1(xn−1) + ηn−1) (4.91) Ne i=1 1 Ne X f2,(i) = xn (4.92) Ne i=1 and,

−1 ˆ f2 T f2 T Kfn = Pn Hn (HnPn Hn + Rn) (4.93)

Deﬁning,

T f2 f2 f2 Pn = Ln GLn (4.94)

where,

f2 f2 Ln = Xn T (4.95) 63 the analysis error covariance matrix will be approximated as:

ˆa ˆ f2 Pn = (I − KfnHn)Pn (4.96)

−1 f2 T f2 T f2 = (I − Pn Hn (HnPn Hn + Rn) Hn)Pn (4.97)

T T −1 T f2 f2 T f2 f2 T f2 f2 = (I − Ln GLn Hn (HnLn GLn Hn + Rn) Hn)Ln GLn (4.98)

h T T −1 i T f2 f2 f2 f2 f2 f2 = Ln G − G(HnLn ) (HnLn G(HnLn ) + Rn) HnLn G Ln (4.99)

h T i−1 T f2 −1 f2 −1 f2 f2 = Ln G + (HnLn ) Rn (HnLn ) Ln (4.100)

T f2 ˆ a f2 = Ln Un Ln (4.101)

ˆ and the Kalman gain Kfn can be expressed as:

T ˆ f2 ˆ a f2 −1 Kfn = Ln Un (HnLn ) Rn (4.102)

In summary, after updating the mean state according to:

T a f2 f2 ˆ a f2 −1 f2 xn = xn + Ln Un (HnLn ) Rn (yn − Hnxn ) (4.103)

The analysis state ensemble is generated using:

−1T a,i a p f2 ˆa T xn = xn + Ne − 1Ln (Cn) Ωn(:,i) (4.104)

−1 ˆa ˆ a where Cn is a square-root of (Un ) such that:

−1 T ˆ a ˆa ˆa (Un ) = CnCn (4.105) which is similar to the form of the standard analysis scheme.

4.4 Summary of the SEIK-OSA algorithms 64

a,(i) Ne • Smoothing Step (Starting from {xn−1}i=1)

N f1,(i) e – Generate the forecast members {xn }i=1 as:

f1,(i) a,(i) (i) (i) xn = Mn−1(xn−1) + ηn−1; ηn−1 ∼ N (0,Qn−1)

– Compute the smoothing mean

T s a a s f1 −1 f1 xn−1 = xn−1 + Ln−1Un−1(HnLn ) Rn (yn − Hnxn )

– Resample the smoothing members √ s,i s a s −1T T xn = xn + Ne − 1Ln−1(Cn−1) Ωn(:,i). T s s T −1 f1 −1 f1 s −1 where, Cn−1Cn−1 = G + (HnLn ) Rn HnLn = (Un−1)

s,(i) Ne • analysis Step (Starting from {xn−1}i=1)

f1 1. Qn−1 = αPn (SEIK-OSAQˆ)

N f2,(i) e – Generate the members {xn }i=1 as:

f2,(i) s,(i) (i) (i) xn = Mn−1(xn−1) + ηn−1; ηn−1 ∼ N (0,Qn−1)

– Compute the analysis mean

−1 a f2 f1 T f1 T f1 xn = xn + αP Hn (HnPn Hn + Rn) (yn − Hnxn )

– Resample the analysis members √ a,i a f1 a T xn = xn + Ne − 1Ln CfnΩn(:,i) T a a 1 2 where CfnCfn = Sn + αSn

2. P f2 = 0 (SEIK-OSA) xn ,εn N f2,(i) e – Generate the members {xn }i=1 as:

f2,(i) s,(i) (i) (i) xn = Mn−1(xn−1) + ηn−1; ηn−1 ∼ N (0,Qn−1)

– Compute the analysis mean

T a f2 f2 ˆ a f2 −1 f2 xn = xn + Ln Un (HnLn ) Rn (yn − Hnxn )

– Resample the analysis members 65 √ −1T a,i a f2 â T xn = xn + Ne − 1Ln (Cn) Ωn(:,i) T T −1 â â −1 f2 −1 f2 ˆ a where CnCn = G + (HnLn ) Rn (HnLn ) = (Un )

4.5 Conclusion

In this chapter, we proposed two OSA-based SEIK algorithms using two different ways to avoid the use of the model error covariance matrix during the analysis step. This led to two different algorithms: the first scheme expresses Q as a fraction of the forecast error covariance matrix where the second one reformulates the analysis equations so that the Q is implicitly accounted for in the equations. In the next chapter, numerical results of these two OSA-based square-root schemes and the stochastic version will be presented and compared to those of the corresponding standard schemes. 66

Chapter 5

Numerical Experiments & Results

5.1 Introduction

Before testing the proposed schemes with a realistic setting of the Advanced Circu- lation model, ADCIRC, configured for a storm surge forecasting problem, the filters are first evaluated using the strongly-nonlinear Lorenz-96 model (L96), a widely used model in the data assimilation community, through an extensive set of sensitivity experiments. We note that the computational cost of the OSA-based EnKFs, is prac- tically double than that of an EnKF since these require 2Ne model runs and 2Ne Kalman corrections. Indeed, as presented in the previous chapters, the OSA-based filters smooth the state at the previous time step and then integrate the model again before updating the state at the current time. This might not be practical in large scale applications. The idea is then to assess whether with comparable computational cost, the OSA-based filters can outperform the standard ones; that is to see if, with half the number of ensemble members, the OSA-based EnKFs can outperform the standard EnKFs.

5.2 Numerical experiments with the Lorenz-96 model

5.2.1 Experimental setting

To evaluate and compare the behavior of the proposed OSA-based EnKF and SEIK schemes and to assess their performances with respect to the standard EnKF and SEIK ﬁlters, numerical experiments were performed with the Lorenz-96 model [38]. 67 The model simulates the time evolution of an atmospheric quantity following a set of diﬀerential equations:

dx i = (x − x ) x − x + F, i = 1, ··· ,N (5.1) dt i+1 i−2 i−1 i x where the nonlinear (quadratic) terms stand for advection and the linear term simulates dissipation. L96 obeys also to the energy conservation law and is sensitive to initial conditions and external forcing. In its most common form, the system dimension is Nx = 40 and the forcing term F is equal to 8 . For this value of F , disturbances propagate from low to high indices (west to east), and the model exhibits chaotic behavior. Boundary conditions are periodic, i.e., x−1 = x39, x0 = x40 and x41 = x1. L96 was discretized using the Runge-Kutta fourth-order scheme with a constant time step ∆t = 0.05 (which corresponds to six hours in real-world time). The trajectory of a reference run is taken as the “true” trajectory to evaluate the filters performances by assessing how well they recover these reference states using a forecast model with perturbed initial conditions. Observations are extracted from the reference states and independently perturbed with Gaussian noise of zero mean and unit variance. Accordingly, the observational error covariance, R, was set to the identity matrix. Different frequencies of assimilated data are considered, ranging from every time step to every 20 time steps. The goal is to test more realistic and challenging situations to test the filtering schemes. Three observational scenarios are considered: full density (i.e., all model state variables are observed), half density (i.e., every second variable is observed) and quarter density (i.e., every fourth variable is observed). In order to generate the filters’ initial forecast ensemble, the model was numerically integrated forward without assimilation for several years in real-world time. The mean of the initial ensemble is set as the mean of this long model run and then the initial ensemble is generated by adding Gaussian noise with zero mean and 68 identity covariance. This choice of the initial ensemble members ensures that they are off the model attractor and do not have a clear relationship between spread and error. A long-enough spin-up period was also considered to remove any detrimental impact. All filters were implemented with the covariance inflation and localization techniques. Since local analysis is scheme independent; that is, it can be implemented with stochastic and deterministic filters, we consider it in our experiments with all the filters. This type of localization restricts the update of each grid point to consider only observations within some influence radius. After a spin-up period of roughly 20 days, simulations were carried over a period of five years (or 7300 model steps). The root-mean-square error (RMSE) between the reference states and the filters’ analyses averaged over all variables and over the assimilation period is used to evaluate the performances of the filters. Given a set of

T n-dimensional state vectors {xk : xk = (xk,1, ··· , xk,n) , k = 1, ··· , kmax}, with kmax being the maximum time index, the time-average RMSE is deﬁned as:

v kmax u n 1 X u 1 X a 2 RMSE\ = t (ˆxk,i − xk,i) , (5.2) kmax n k=1 i=1

a a a T wherex ˆk = (ˆxk,1, ··· , xˆk,n) is the analysis at time k. All experiments are independently repeated L = 10 times each time with a randomly generated initial ensemble and observational errors. The average of the RMSEs of these L = 10 runs deﬁned as:

L 1 X RMSE = RMSE\ (5.3) L l l=1 is taken as the ﬁnal result to reduce statistical ﬂuctuations. To reproduce realistic scenarios, model errors were not accounted for in the forecast model in all experiments unless otherwise mentioned. 69 5.2.2 Results and discussion

The OSA-based filters are tested through a series of sensitivity experiments with different ensemble sizes and spatial and temporal observation frequencies to assess their behavior under different experimental settings. The results will be presented using two-dimensional plots of the RMSE as a function of the inflation factor and the localization scales. Localization support radii will vary between 2 (strong localization) to 40 (weak localization) grid points. Inflation will vary from 1 to 1.3 when no bias is introduced to the model and from 1 to 2.5 if we run a biased model.

The SEIK-OSAQˆ filter presented in Chapter 4 which overcomes the problem of the presence of the model error covariance Q in the analysis equations by a simple approximation of this matrix is first tested. Then, a sensitivity study will be conducted to assess the benefit of the SEIK-OSA algorithm. We will also study the performance of the EnKF-OSA

SEIK-OSAQˆ

The SEIK-OSAQˆ ﬁlter requires tuning the parameter α, the fraction between the model error covariance and the forecast error covariance under the approximation

f1 Qn−1 = αPn , which is done here by trial and error. In the first set of experiments, we fix the ensemble size to Ne = 10 and study the sensitivity of the filter to the number of observations. The optimal values of α, in terms of RMSE, when all, half and quarter of the observations are assimilated were found to be equal to 0.1, 0.1 and 0.2, respectively. Figure (5.1) shows the performances of SEIK-Reg and SEIK-

OSAQˆ ﬁlters for the three observation scenarios. Using SEIK-Reg, many divergences occur when only half or quarter of the observations are assimilated. In contrast, The

SEIK-OSAQˆ ﬁlter is much more robust robust. It is also noticeable that, when all observations are assimilated, the SEIK-OSAQˆ performs better than SEIK-Reg. When only half of the observations are assimilated, the two schemes produce comparable 70

Seik-OSA , All Obs Seik-Reg, All Obs Qapprox Min = 0.44 Min = 0.41 1.5 1.5 1.25 1.25 1.2 1.2 1.15 1.15 1.1 1.1 0.5 0.5 1.05 1.05 Inflation 1 0.35 1 0.35 10 20 30 40 10 20 30 40

Seik-OSA , Half Obs Seik-Reg, Half Obs Qapprox Min = 0.84 Min = 0.83 3 3 1.25 1.25 1.2 1.2 1.15 1.15 1.1 1.1 1 1 1.05 1.05 Inflation 1 0.7 1 0.7 10 20 30 40 10 20 30 40

Seik-OSA , Quarter Obs Seik-Reg, Quarter Obs Qapprox Min = 1.52 Min = 1.65 4 4 1.25 1.25 1.2 1.2 1.15 1.15 2 2 1.1 1.1 1.05 1.05 Inflation 1 1.15 1 1.15 10 20 30 40 10 20 30 40 Localization Localization

Figure 5.1: Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis). The two filters are implemented with 10 members and assimilation of observations from (top) all model variables (middle) half and (bottom) quarter of the variables at every 4 model time steps (or 24 h in real time). A logarithmic color scale is used to emphasize the low RMSE values. The minimum- averaged RMSEs are indicated by asterisks, and their associated values are given in the title. White boxes indicate divergence of the filter.

results. However, when quarter of the observations are assimilated, SEIK-Reg out-

performs SEIK-OSAQˆ. To further assess the behavior of the proposed scheme for diﬀerent numbers of observations, we repeated the same experiment as in Figure (5.1), but using 20 members instead of 10 with the same values of α. The results are then reported in Figure (5.2). The results suggest that even when the ensemble size increases, the SEIK-

OSAQˆ filter performs only slightly better than the SEIK-Reg when all and half of the observations are assimilated. But, again, when only quarter of the observations are assimilated, the standard SEIK outperforms SEIK-OSAQˆ filter. In the third set of experiments, we test the proposed scheme with higher frequencies of observations. To do so, we fix the ensemble size to 20 and we consider only the case where all the observations are assimilated. Assimilations are performed every 1, 2 and 4 model steps 71

Seik-OSA , All Obs Seik-Reg, All Obs Qapprox Min = 0.40 Min = 0.38 1 1 1.25 1.25 1.2 1.2

1.15 0.6 1.15 0.6 1.1 1.1 1.05 1.05

Inflation 0.4 0.4 1 0.35 1 0.35 10 20 30 40 10 20 30 40

Seik-OSA , Half Obs Seik-Reg, Half Obs Qapprox Min = 0.72 Min = 0.70 2 2 1.25 1.25 1.2 1.2 1.15 1.15 1.1 1 1.1 1 1.05 1.05 Inflation 1 0.62 1 0.62 10 20 30 40 10 20 30 40

Seik-OSA , Quarter Obs Seik-Reg, Quarter Obs Qapprox Min = 1.18 Min = 1.24 3 3 1.25 1.25 1.2 1.2 1.15 1.15 1.1 1.1 1.05 1.05 Inflation 1 1 1 1 10 20 30 40 10 20 30 40 Localization Localization

Figure 5.2: Same as Fig. (5.1), but for 20 ensemble members. with values of α respectively equal to 0.01, 0.05 and 0.1. The results are reported in

Figure (5.3). In all cases, SEIK-OSAQˆ ﬁlter is either similar or slightly better than

Seik-OSA , All Obs Seik-Reg, All Obs Qapprox Min = 0.20 Min = 0.20 0.6 0.6 1.25 1.25 1.2 1.2 1.15 1.15 1.1 1.1 1.05 1.05 Inflation 1 0.2 1 0.2 10 20 30 40 10 20 30 40

Seik-OSA , All Obs Seik-Reg, All Obs Qapprox Min = 0.29 Min = 0.28 0.6 0.6 1.25 1.25 1.2 1.2 1.15 1.15 0.4 0.4 1.1 1.1 1.05 1.05 Inflation 0.3 0.3 1 0.28 1 0.28 10 20 30 40 10 20 30 40

Seik-OSA , All Obs Seik-Reg, All Obs Qapprox Min = 0.40 Min = 0.38 1 1 1.25 1.25 1.2 1.2

1.15 0.6 1.15 0.6 1.1 1.1 1.05 1.05

Inflation 0.4 0.4 1 0.35 1 0.35 10 20 30 40 10 20 30 40 Localization Localization

Figure 5.3: Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis). The two filters are implemented with 20 members and assimilation of observations every (top) 1 model time step (middle) 2 and (bottom) 4 model time steps. A logarithmic color scale is used to emphasize the low RMSE values. 72 SEIK-Reg. However, the results are statistically not very significant. In terms of

robustness to the choice of localization and inflation, SEIK-OSAQˆ filter is found to be more robust especially with small inflation factors and weak localization. This

might be explained by the fact that the approximation on which the SEIK-OSAQˆ filter is built, is basically an inflation of the covariance matrix by a factor α. When the inflation factor is small, the inflation coming from the approximation makes the

estimates more accurate. SEIK-OSAQˆ filter was not shown to significantly improve the results of SEIK-Reg. A possible reason that might explain this could be related to the approximation itself. In fact, when the model is perfect (no bias or model errors are considered), Q is assumed to be zero. Thus, the value assigned to α should be small for consistency. However, assigning a small value to α, reduces the effect of the second KF correction (i.e. analysis step) and consequently, the contribution of the OSA-based formulation becomes non-significant. We thus, investigate the robustness of the proposed scheme to bias and model errors. The OSA-based schemes are also expected to produce a better background, and thus, one may expect that, in the presence of model errors, the OSA-based filtering strategy should be more efficient than the standard filtering scheme.

The SEIK-OSAQˆ ﬁlter was then tested and compared to SEIK-Reg after introducing model errors in L96. This was done by setting the forcing F = 6 in equation (5.1) in the forecast model, while the reference states were forced with F = 8. Figure (5.4) presents the results of these simulations. The value of α was tuned to 0.3 in all

observation scenarios. Figure (5.4) shows that SEIK-OSAQˆ is beneﬁcial only when all observations are assimilated. As in the perfect case, when the number of observations decreases to half or quarter, the standard scheme outperforms the proposed OSA- based algorithm. A series of experiments where model errors, diﬀerent from zero, were added to generate the reference states and then accounted for in the forecast model, were also conducted and, again, no improvements were noticeable in terms 73

Seik-OSA , All Obs Seik-Reg, All Obs Qapprox Min = 0.66 Min = 0.64 2 2 2.5 2.5

2 2

1 1 1.5 1.5 Inflation 1 0.64 1 0.64 10 20 30 40 10 20 30 40

Seik-OSA , Half Obs Seik-Reg, Half Obs Qapprox Min = 1.11 Min = 1.16 3 3 2.5 2.5

2 2

1.5 1.5 Inflation 1 1 1 1 10 20 30 40 10 20 30 40

Seik-OSA , Quarter Obs Seik-Reg, Quarter Obs Qapprox Min = 1.84 Min = 2.02 4 4 2.5 2.5

2 2

1.5 2 1.5 2 Inflation 1 1.53 1 1.53 10 20 30 40 10 20 30 40 Localization Localization

Figure 5.4: Same as Fig. (5.1), but for 20 ensemble members and F = 6 (instead of 8) in ﬁlters.

f1 of RMSE. This suggests that the approximation Qn−1 = αPn is not suitable in the tested cases.

EnKF-OSA VS SEIK-OSA

This section will evaluate the performance of the SEIK-OSA scheme and compare them to those of SEIK-Reg and EnKF-OSA.

Sensitivity to ensemble size

We first study the sensitivity of the four ffilters (EnKF-Reg, EnKF-OSA, SEIK-Reg and SEIK-OSA) to the ensemble size, Ne. In realistic large-scale applications, one would be restricted by computational resources to using small ensembles. This is why, computing accurate state estimates with small ensembles is very desirable. The experiments are conducted using three ensemble sizes: Ne = 10, 20, 40. We fix the frequency of the observations to one day, that is, assimilations are performed every four model steps. We consider the three observational scenarios where all, half and quarter 74 of the observations are assimilated. Figures (5.5), (5.6) and (5.7) present the perfor-

EnKF-Reg, All Obs EnKF-OSA, All Obs SEIK-Reg, All Obs SEIK-OSA, All Obs Min = 0.52 Min = 0.46 Min = 0.44 Min = 0.38 1.5 1.5 1.5 1.5 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 1.1 1.1 1.1 0.5 0.5 0.5 0.5 Inflation Inflation 1 0.35 1 0.35 1 0.35 1 0.35 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Half Obs EnKF-OSA, Half Obs SEIK-Reg, Half Obs SEIK-OSA, Half Obs Min = 1.06 Min = 0.87 Min = 0.84 Min = 0.70 3 3 3 3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 1.1 1.1 1.1 1 1 1 1 Inflation Inflation 1 0.7 1 0.7 1 0.7 1 0.7 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Quarter Obs EnKF-OSA, Quarter Obs SEIK-Reg, Quarter Obs SEIK-OSA, Quarter Obs Min = 1.76 Min = 1.40 Min = 1.52 Min = 1.18 4 4 4 4 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 2 2 2 2 1.1 1.1 1.1 1.1 Inflation Inflation 1 1.15 1 1.15 1 1.15 1 1.15 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.5: Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis). All filters are implemented with 10 members and assimilation of observations from (top) all model variables (middle) half and (bottom) quarter of the variables at every 4 model time steps (or 24 h in real time). A logarithmic color scale is used to emphasize the low RMSE values. The minimum-averaged RMSEs are indicated by asterisks, and their associated values are given in the title. White boxes indicate divergence of the filter. mances of the standard and OSA-based schemes in terms of RMSE for both EnKF and SEIK with Ne = 10, 20 and 40, respectively. In the standard filtering scheme, it is known that the deterministic filters outperform the stochastic EnKF when the number of independent observations is larger than the ensemble size [10]. This is consistent with our assimilation results. In fact, going from Ne = 10 to Ne = 40, the difference in terms of the minimum RMSE between SEIK and EnKF becomes smaller whether filtering is implemented following the OSA formulation or not. For instance, consider the case where half of the observations are assimilated. When only 10 members are used, both SEIK-Reg and SEIK-OSA achieve a better RMSE minimum than EnKF-Reg and EnKF-OSA, with a percentage of improvement around 20% for both. With 40 members, however, the difference between the square-root filters and the 75

EnKF-Reg, All Obs EnKF-OSA, All Obs SEIK-Reg, All Obs SEIK-OSA, All Obs Min = 0.44 Min = 0.39 Min = 0.40 Min = 0.35 1 1 1 1 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 0.6 0.6 0.6 0.6

1.1 1.1 1.1 1.1

Inflation 0.4 0.4 Inflation 0.4 0.4 1 0.35 1 0.35 1 0.35 1 0.35 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Half Obs EnKF-OSA, Half Obs SEIK-Reg, Half Obs SEIK-OSA, Half Obs Min = 0.79 Min = 0.69 Min = 0.72 Min = 0.62 2 2 2 2 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1 1 1 1 1.1 1.1 1.1 1.1 Inflation Inflation 1 0.62 1 0.62 1 0.62 1 0.62 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Quarter Obs EnKF-OSA, Quarter Obs SEIK-Reg, Quarter Obs SEIK-OSA, Quarter Obs Min = 1.26 Min = 1.07 Min = 1.18 Min = 0.99 3 3 3 3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 1.1 1.1 1.1 Inflation Inflation 1 1 1 1 1 1 1 1 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.6: Same as Fig. (5.5), but for 20 ensemble members.

stochastic ones becomes less significant especially for the OSA-based scheme with the percentage of improvement less than 2%. As explained previously, this is related to the fact that the stochastic EnKF suffers from the observational error undersampling problem when a small ensemble size is considered. The performances of EnKF-OSA and SEIK-OSA are clearly better than those of EnKF-Reg and SEIK-Reg for all ensemble sizes. Consider first the most challenging

case where Ne = 10. Figure (5.5) shows that for the stochastic version, the minimum average RMSEs achieved by EnKF-OSA are respectively 0.46 for the experiment assimilating all model variables, 0.87 for the experiment assimilating half of the model variables, and 1.4 for the experiment assimilating only quarter of the model variables, compared to respectively, 0.52 ,1.06 and 1.76 for EnKF-Reg. SEIK also exhibits similar behavior but with smaller error values when the considered ensemble size is small. SEIK-OSA, in turn, achieves a minimum RMSE of 0.38, 0.7 and 1.18 when all, half and quarter of the observations are assimilated, respectively, compared to 0.44, 0.84 and 1.52 with EnKF-Reg. In other words, the beneﬁt of using the OSA-based ﬁlters 76

EnKF-Reg, All Obs EnKF-OSA, All Obs SEIK-Reg, All Obs SEIK-OSA, All Obs Min = 0.40 Min = 0.35 Min = 0.38 Min = 0.34 0.8 0.8 0.8 0.8 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 1.1 1.1 1.1 0.4 0.4 0.4 0.4 Inflation Inflation 1 0.34 1 0.34 1 0.34 1 0.34 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Half Obs EnKF-OSA, Half Obs SEIK-Reg, Half Obs SEIK-OSA, Half Obs Min = 0.70 Min = 0.61 Min = 0.66 Min = 0.60 1.5 1.5 1.5 1.5 1.3 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1 1 1 1

1.1 0.8 1.1 0.8 1.1 0.8 1.1 0.8 Inflation Inflation 1 0.6 1 0.6 1 0.6 1 0.6 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Quarter Obs EnKF-OSA, Quarter Obs SEIK-Reg, Quarter Obs SEIK-OSA, Quarter Obs Min = 1.05 Min = 0.92 Min = 1.01 Min = 0.92 2.5 2.5 2.5 2.5 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 1.5 1.5 1.5 1.5 1.1 1.1 1.1 1.1

Inflation 1 1 Inflation 1 1 1 0.92 1 0.92 1 0.92 1 0.92 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.7: Same as Fig. (5.5), but for 40 ensemble members.

is accentuated when few data are assimilated. Moreover, using small ensemble mem-

bers (in particular Ne = 10) and assimilating less data clearly affects the robustness of the EnKF-Reg and SEIK-Reg with respect to the choice of localization and inflation. When assimilating only half or quarter of the observations, the standard filters’ performance degrades and divergences (indicated by the white boxes in the figure) occur. In contrast, the OSA-based filters exhibit very robust behavior with respect to the choice of localization and inflation. When tested with relatively larger ensemble

sizes, Ne = 20 and Ne = 40, both EnKF-OSA and SEIK-OSA still perform better than EnKF-Reg and SEIK-Reg in all observational densities scenarios. The percentages of improvement provided by the OSA-based algorithms in terms of estimation accuracy are reported in Table 5.1 for Ne = 10, 20 and 40 and for all the observational scenarios. We notice that the improvement introduced by the SEIK-OSA ﬁlter is, in general, better than that introduced by EnKF-OSA when the

ensemble size is small (particularly Ne = 10, 20). With 40 members, however, EnKF- OSA algorithm shows a better improvement than SEIK-OSA, as expected. 77 EnKF-OSA SEIK-OSA Ne = 10 Ne = 20 Ne = 40 Ne = 10 Ne = 20 Ne = 40 all 12.53 11.18 10.64 13.64 11.68 10.02 half 18.02 13.13 12.10 16.48 13.4 8.52 quarter 20.17 15.53 11.80 22 16.3 8.54

Table 5.1: Best percentages of improvement, in terms of RMSE, introduced by the OSA-based ﬁlters compared to the standard ﬁlters using the same ensemble size.

Table 5.1 shows also that, with only 10 or 20 members, the improvement in estimation accuracy by the OSA-based ﬁlters increases as the number of observations decreases. This is desirable and suitable for real applications where the ensemble size and the number of observations are often limited. To further assess the behavior of the OSA-based schemes with large ensembles, we

performed, under the same setup, new experiments with Ne = 80 and 160. For

these two cases where Ne is larger than the state dimension, no localization is conducted. Figure (5.8) presents bar plots of the minimum RMSE resulting from the

Min RMSE Min RMSE Min RMSE 1.8 0.6 EnKF−Reg EnKF−Reg EnKF−Reg EnKF−OSA 1.1 EnKF−OSA EnKF−OSA 0.55 1.6 SEIK−Reg SEIK−Reg SEIK−Reg SEIK−OSA 1 SEIK−OSA SEIK−OSA 0.5 1.4 0.9 0.45 0.8 1.2 0.4 0.7 1 0.35 0.6 0.8 0.3 0.5

0.25 0.4 0.6 Ne = 10 Ne = 20 Ne = 40 Ne = 80 Ne = 160 Ne = 10 Ne = 20 Ne = 40 Ne = 80 Ne = 160 Ne = 10 Ne = 20 Ne = 40 Ne = 80 Ne = 160 (a) All observations (b) Half observations (c) Quarter observations

Figure 5.8: Minimum average RMSE for all tested ﬁlters (EnKF-Reg, EnKF-OSA, SEIK-Reg and SEIK-OSA) as a function of the ensemble size. left (All) middle (half) and right (quarter) observations are observed every 4 model time steps.

separately optimized conﬁguration of each ﬁlter, as a function of the ensemble size

Ne = 10, 20, 40, 80 and 160 and for full, half and quarter observation scenarios. Over- all, when changing the ensemble size, EnKF-OSA and SEIK-OSA outperform the EnKF-Reg and SEIK-Reg. The OSA-based ﬁlters lead to the best improvements when the ensemble size is small, especially when the number of assimilated observa- 78 EnKF-OSA SEIK-OSA Ne/2Ne 10/20 20/40 40/80 80/160 10/20 20/40 40/80 80/160 all -4.08 1.98 6.65 5.48 4.64 7.01 2.4 8.64 half -9.38 1.8 9.14 5.22 2.22 5.67 4.47 3.14 quarter -10.94 -1.63 9.18 0.98 0.01 1.13 8.54 4.56

Table 5.2: Best percentages of improvement, in terms of RMSE, introduced by the OSA-based filters compared to the standard filters using only half of the ensemble size. The (-) sign means that the standard filter with 2Ne produces better estimates than the OSA-based filter with Ne. tions is small. Starting from 80 members, however, SEIK-OSA performances level off, except for the case when quarter observations are assimilated. In contrast, the stochastic EnKF-OSA continues improving as the ensemble size increases. Figure (5.8) also reveals an important feature of the OSA-based filters. These latter are not only outperforming the corresponding standard schemes with the same ensemble size, but, in most of the cases, the proposed OSA-based schemes outperform the standard ones using only half of the members. One effective way to further evaluate this is to compute the percentage of improvement introduced by the OSA-based filters using only half of the number of members used in the standard ones. These percentages are reported in Table 5.2. The notation Ne/2Ne means that the OSA filter uses Ne members and the standard filter uses 2Ne members. Table 5.2 illustrates that, with small members (10/20), only SEIK-OSA outperforms SEIK-Reg. How- ever, with 20/40 members, EnKF-OSA starts introducing improvement compared to EnKF-Reg if all or half of the observations are assimilated, while for SEIK-OSA, the percentage of improvement increases with the three observation scenarios. With large ensembles, both OSA-based filters can outperform the corresponding standard filters with only half of the members. This suggests that SEIK-OSA is more beneficial than EnKF-OSA with small ensembles. Overall, in all tested cases, SEIK-OSA was found to perform better than SEIK-Reg for the same computational cost. 79 Sensitivity to the frequency of observations

In the second set of experiments, we fix the ensemble size to Ne = 20 and change only the temporal frequency of observations, i.e., the ratio between the analysis step and the model step. Let FreqObs denote this ratio. We test the filters with nine different values of FreqObs = 1, 2, 4, 6, 8, 10, 12, 16, 20 to reproduce the situations in which the observations are available and frequent in time as well as the challenging situations in which data are not frequent. Figures (5.9), (5.10) and (5.11) show the results of these experiments when observations are respectively assimilated every 1, 2 and 6 model steps. Figure (5.9) illustrates that, when the observations are very fre-

EnKF-Reg, All Obs EnKF-OSA, All Obs SEIK-Reg, All Obs SEIK-OSA, All Obs Min = 0.23 Min = 0.23 Min = 0.20 Min = 0.20 0.6 0.6 0.6 0.6 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 1.1 1.1 1.1 Inflation Inflation 1 0.2 1 0.2 1 0.2 1 0.2 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Half Obs EnKF-OSA, Half Obs SEIK-Reg, Half Obs SEIK-OSA, Half Obs Min = 0.39 Min = 0.37 Min = 0.34 Min = 0.34 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3

0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 0.6 0.6 0.6 0.6 1.1 1.1 1.1 1.1

Inflation 0.4 0.4 Inflation 0.4 0.4 1 0.34 1 0.34 1 0.34 1 0.34 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Quarter Obs EnKF-OSA, Quarter Obs SEIK-Reg, Quarter Obs SEIK-OSA, Quarter Obs Min = 0.60 Min = 0.60 Min = 0.52 Min = 0.52 3 3 3 3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1 1 1 1 1.1 1.1 1.1 1.1 Inflation Inflation 1 0.52 1 0.52 1 0.52 1 0.52 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.9: Same as Fig. (5.5), but for 20 ensemble members and assimilation every model time step. quent in time (assimilated at every model step), both OSA-based ﬁlters and standard ones produce similar results. No improvement is introduced by the OSA variants in this case. When the observations are assimilated frequently, the beneﬁt of the smoothing step becomes negligible and the performance of the OSA-based scheme will be similar to the standard one. Indeed, when the frequency of assimilation cy- 80

EnKF-Reg, All Obs EnKF-OSA, All Obs SEIK-Reg, All Obs SEIK-OSA, All Obs Min = 0.32 Min = 0.31 Min = 0.29 Min = 0.28 0.6 0.6 0.6 0.6 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 0.4 0.4 0.4 0.4 1.1 1.1 1.1 1.1

Inflation 0.3 0.3 Inflation 0.3 0.3 1 0.28 1 0.28 1 0.28 1 0.28 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Half Obs EnKF-OSA, Half Obs SEIK-Reg, Half Obs SEIK-OSA, Half Obs Min = 0.54 Min = 0.51 Min = 0.48 Min = 0.46 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3

1.2 0.8 1.2 0.8 1.2 0.8 1.2 0.8

1.1 0.6 1.1 0.6 1.1 0.6 1.1 0.6 Inflation Inflation 1 0.46 1 0.46 1 0.46 1 0.46 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Quarter Obs EnKF-OSA, Quarter Obs SEIK-Reg, Quarter Obs SEIK-OSA, Quarter Obs Min = 0.84 Min = 0.78 Min = 0.76 Min = 0.73 3 3 3 3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 1.1 1.1 1.1 1 1 1 1 Inflation Inflation 1 0.73 1 0.73 1 0.73 1 0.73 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.10: Same as Fig. (5.5), but for 20 ensemble members and assimilation every two model time steps. cles is high, the model becomes weakly nonlinear within the forecast period and the algorithms behave like the KF, especially when the ensemble size is not small. When the observational frequency is lower, a situation for which the descretized model becomes more nonlinear, the EnKF-OSA and SEIK-OSA ﬁlters start producing more accurate results than EnKF-Reg and SEIK-Reg. For example, Figure (5.10) shows that the performances of the OSA-based ﬁlters are slightly better than those of the standard ones when the assimilation is performed every two model steps. With lower frequencies of observations, (every four or six model steps), the improvements of the OSA-based schemes become clearly pronounced. In the case where observations are assimilated every six model steps, the performances of the proposed EnKF-OSA and SEIK-OSA, as seen from Figure (5.11), are robust and result in more accurate estimates compared to those obtained by EnKF-Reg and SEIK-Reg. The EnKF-OSA leads to about 15%, 20% and 16% improvement over EnKF-Reg when all, half and quarter of the observations are assimilated, respectively. The SEIK-OSA, in turn, leads to 18%, 22% and 21% improvement over the SEIK-Reg considering the same 81

EnKF-Reg, All Obs EnKF-OSA, All Obs SEIK-Reg, All Obs SEIK-OSA, All Obs Min = 0.54 Min = 0.46 Min = 0.50 Min = 0.41 2 2 2 2 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 1 1 1 1

1.1 1.1 1.1 1.1

Inflation 0.5 0.5 Inflation 0.5 0.5 1 0.41 1 0.41 1 0.41 1 0.41 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Half Obs EnKF-OSA, Half Obs SEIK-Reg, Half Obs SEIK-OSA, Half Obs Min = 1.17 Min = 0.94 Min = 1.11 Min = 0.87 3 3 3 3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 1.1 1.1 1.1

Inflation 1 1 Inflation 1 1 1 0.87 1 0.87 1 0.87 1 0.87 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF-Reg, Quarter Obs EnKF-OSA, Quarter Obs SEIK-Reg, Quarter Obs SEIK-OSA, Quarter Obs Min = 1.74 Min = 1.47 Min = 1.73 Min = 1.37 4 4 4 4 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2

1.1 2 1.1 2 1.1 2 1.1 2 Inflation Inflation 1 1.37 1 1.37 1 1.37 1 1.37 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.11: Same as Fig. (5.5), but for 20 ensemble members and assimilation every six model time steps.

observation scenarios. We further study the behavior of the OSA-based ﬁlters when the frequency of observations is lower by conducting a set of experiments considering the cases where the observations are assimilated every 8, 10, 12, 16 and 20 model steps. Figure (5.12)

1 3 3 E nKF − Reg 0.9 E nKF − OSA − 2.5 SEIK Reg 2.5 0.8 SEIK − OSA

2 0.7 2

0.6 1.5

1.5 0.5 Min rmse Min rmse Min rmse 1

0.4 1 0.5 0.3

0.2 0 0.5 1 2 4 6 8 10 12 14 16 20 1 2 4 6 8 10 12 14 16 20 1 2 4 6 8 10 12 14 16 20 FreqObs FreqObs FreqObs (a) All observations (b) Half observations (c) Quarter observations

Figure 5.12: Minimum average RMSE for all tested filters (EnKF-Reg, EnKF-OSA, SEIK-Reg and SEIK-OSA) as a function of the frequency of observations. (left) All (middle) half and (right) quarter observations are observed every every 4 model steps with Ne=20. illustrates the evolution of the minimum RMSE as the frequency of observations changes for the standard and OSA-based schemes. An important common feature is 82 revealed from this figure for all the observation frequencies. For very high and very low frequencies, the OSA-based and standard schemes produce similar results. For frequencies of observations ranging from two to twelve model steps, the improvements of the OSA-based filters over the standard schemes are more pronounced. For lower frequencies, these improvements level off. The largest improvements are obtained when the observations are assimilated every six model steps, then, this becomes less important as the time frequency of available observations decreases until stabilizing after a certain frequencies. This can be explained by the fact that, by construction, the OSA-based schemes apply a one-step-ahead smoothing step to the analyzed ensemble members. Therefore, the more data available in time, the greater the number of smoothing steps applied, and hence the better the characterization of the state. In other words, the smoothing step of the state ensemble is expected to enhance its statistics and to provide more consistent cross-correlations. This should improve the prediction of the data at the next analysis step. These results demonstrate not only the effectiveness, but also the robustness of the OSA-based filtering schemes with respect to the frequencies of available observations.

Sensitivity to measurement errors

In this set of experiments, the ensemble size is set to Ne = 20 and the observations are assimilated every four model steps. To study the sensitivity of the proposed OSA-based schemes to observational errors, two new values of the observational error variance, a relatively small value, 0.1, and a relatively large value, 2, were tested. Figures (5.13) and (5.14) illustrate the results of these experiments. When comparing these results with those of Figure (5.6), where the measurement error variance is equal to 1, one can see, as expected, that all the ﬁlters’ performances degrade as the noise level in the observations increases. The ﬁlters exhibit similar performances in the case of small observational errors. 83

EnKF−Reg, All Obs EnKF−OSA, All Obs Min = 0.12 Min = 0.12 0.8 0.8 1.3 1.3

1.2 0.4 1.2 0.4

1.1 0.2 1.1 0.2 Inflation

1 0.11 1 0.11 10 20 30 40 10 20 30 40

EnKF−Reg, Half Obs EnKF−OSA, Half Obs Min = 0.20 Min = 0.19 1.2 1.2 1.3 1.3 0.8 0.8 1.2 0.6 1.2 0.6 0.4 0.4 1.1 1.1 Inflation 0.2 0.2 1 0.17 1 0.17 10 20 30 40 10 20 30 40

EnKF−Reg, Quarter Obs EnKF−OSA, Quarter Obs Min = 0.28 Min = 0.29 2 2 1.3 1.3

1.2 1 1.2 1

1.1 0.5 1.1 0.5 Inflation

1 0.25 1 0.25 10 20 30 40 10 20 30 40 Localization Localization (a) EnKF (b) SEIK

Figure 5.13: Same as Fig. (5.5), but for 20 ensemble members and observational error variance equal to 0.1.

As the noise level in the observations increases, the accuracy of the EnKF-OSA scheme is always better than that of EnKF-Reg. Likewise, SEIK-OSA also outperforms SEIK-Reg for the different observation noise scenarios. For instance, when the observational error is equal to 2, the estimates are approximately 12%, 12% and 8% better when all, half and quarter of the observations are, respectively, assimilated with the EnKF-OSA, and 14%, 11% and 9% better with the SEIK-OSA. For a small variance of measurement error, however, these improvements reduce to only 3%, 2% and −3% with the EnKF-OSA and 4%, 3% and −4% with the SEIK-OSA, suggesting more robustness of the OSA-based-schemes to observation noise. in Figure (5.15), we plot the minimum RMSE values for all the filters considering four values of measurement error variances. We add the case where the signal to noise ratio is high to assess the filters’ behavior in these challenging situations. Overall, the SEIK filter always outperforms the stochastic filter in both standard and OSA-based schemes. Moreover, the improvements introduced by the OSA-based algorithms when 84

EnKF−Reg, All Obs EnKF−OSA, All Obs Min = 0.66 Min = 0.58 2 2 1.3 1.3

1.2 1.2 1 1 1.1 1.1 Inflation

1 0.52 1 0.52 10 20 30 40 10 20 30 40

EnKF−Reg, Half Obs EnKF−OSA, Half Obs Min = 1.20 Min = 1.06 3 3 1.3 1.3

1.2 2 1.2 2

1.5 1.5 1.1 1.1 Inflation

1 0.98 1 0.98 10 20 30 40 10 20 30 40

EnKF−Reg, Quarter Obs EnKF−OSA, Quarter Obs Min = 1.73 Min = 1.59 4 4 1.3 1.3

3 3 1.2 1.2 2.5 2.5

1.1 2 1.1 2 Inflation

1 1.51 1 1.51 10 20 30 40 10 20 30 40 Localization Localization (a) EnKF (b) SEIK

Figure 5.14: Same as Fig. (5.5), but for 20 ensemble members and observational error variance equal to 2.

observations are very noisy (2 and 4) are sensitive to the number of assimilated observations. When this number is large (all observations are assimilated), the diﬀerence between the OSA-based schemes and the standard ones is more pronounced, but, becomes less signiﬁcant with less observations, especially when only quarter of the observations are assimilated.

Sensitivity to bias in the model

This section assesses the results of the OSA-based schemes in the presence of model bias and evaluate their performances with respect to the standard filtering strategies. To introduce model error in L96, the simulations were carried out with F = 6 in equation (5.1) of the forecast model while the reference states were forced with F = 8. In all experiments performed previously, the model bias was not accounted for in the forecast model. Due to the presence of bias in the model, we test the schemes with higher values of inflation factors ranging from 1 to 2.5, considering inflation as a way 85

1 1.8 2.2 E nKF − Reg 0.9 E nKF − OSA 1.6 2 − SEIK Reg 1.8 0.8 SEIK − OSA 1.4 1.6 0.7 1.2

1.4 0.6 1 1.2 0.5 0.8 1 Min rmse Min rmse Min rmse 0.4 0.6 0.8

0.3 0.4 0.6

0.2 0.2 0.4

0.1 0 0.2 0.1 1 2 4 0.1 1 2 4 0.1 1 2 4 Observational error variance Observational error variance Observational error variance

(a) All observations (b) Half observations (c) Quarter observations

Figure 5.15: Minimum average RMSE for all tested filters (EnKF-Reg, EnKF-OSA, SEIK-Reg and SEIK-OSA) as a function of the measurement error variance. (left) All (middle) half and (right) quarter observations are assimilated every 4 model steps with Ne=20. to account for model errors that were neglected in the forecast model. Figure (5.16) shows the RMSE for the different experiments using several values of inflation factor and localization radii. The ensemble size is equal to 20, data is assimilated every four model steps and the observational error variance is set to 1.

EnKF−Reg, All Obs EnKF−OSA, All Obs SEIK−Reg, All Obs SEIK−OSA, All Obs Min = 0.69 Min = 0.67 Min = 0.66 Min = 0.64 2 2 2 2 2.5 2.5 2.5 2.5

2 2 2 2

1 1 1 1 1.5 1.5 1.5 1.5 Inflation Inflation

1 0.64 1 0.64 1 0.64 1 0.64 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF−Reg, Half Obs EnKF−OSA, Half Obs SEIK−Reg, Half Obs SEIK−OSA, Half Obs Min = 1.16 Min = 1.04 Min = 1.11 Min = 1.00 3 3 3 3 2.5 2.5 2.5 2.5

2 2 2 2 2 2 2 2

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 Inflation Inflation

1 1 1 1 1 1 1 1 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF−Reg, Quarter Obs EnKF−OSA, Quarter Obs SEIK−Reg, Quarter Obs SEIK−OSA, Quarter Obs Min = 1.89 Min = 1.60 Min = 1.84 Min = 1.53 4 4 4 4 2.5 2.5 2.5 2.5

3 3 3 3 2 2 2 2 2.5 2.5 2.5 2.5

1.5 2 1.5 2 1.5 2 1.5 2 Inflation Inflation

1 1.53 1 1.53 1 1.53 1 1.53 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.16: Same as Fig. (5.16), but for 20 ensemble members and assimilation every 4 model time steps. Simulations are carried out with an incorrect forcing F = 6.

The results suggest that EnKF-OSA and SEIK-OSA still outperform EnKF-Reg 86 and SEIK-Reg when the model is biased. The relative improvement varies with the observation scenarios. Indeed, as the number of assimilated observations decreases, the improvement introduced by the OSA-based version increases for both stochastic EnKF and deterministic SEIK variants. The EnKF-OSA is 3%, 10% and 15% better than EnKF-Reg when all, half and quarter of the observations are assimilated, respectively, while, SEIK-OSA is 4%, 10% and 17% better than SEIK-Reg. Another experiment is conducted with more frequent observations in time (every 2 model steps) and the results are reported in Figure (5.17). As in the perfect case (i.e. no bias is introduced to the model), when the observations are assimilated more frequently in time, the difference between the OSA-based and the standard schemes becomes less pronounced. However, the main difference with respect to the perfect case scenario is that the difference between the OSA-based filtering strategy and the standard scheme becomes more noteworthy when only quarter of the observations are assimilated.

EnKF−Reg, All Obs EnKF−OSA, All Obs Seik−Reg, All Obs Seik−OSA, All Obs Min = 0.59 Min = 0.57 Min = 0.57 Min = 0.55 0.9 0.9 0.9 0.9 2.5 2.5 2.5 2.5 0.8 0.8 0.8 0.8 2 2 2 2 0.7 0.7 0.7 0.7 1.5 1.5 1.5 1.5

Inflation 0.6 0.6 Inflation 0.6 0.6 1 0.55 1 0.55 1 0.55 1 0.55 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF−Reg, Half Obs EnKF−OSA, Half Obs Seik−Reg, Half Obs Seik−OSA, Half Obs Min = 0.95 Min = 0.89 Min = 0.90 Min = 0.85 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5

2 1.2 2 1.2 2 1.2 2 1.2

1.5 1 1.5 1 1.5 1 1.5 1 Inflation Inflation

1 0.85 1 0.85 1 0.85 1 0.85 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

EnKF−Reg, Quarter Obs EnKF−OSA, Quarter Obs Seik−Reg, Quarter Obs Seik−OSA, Quarter Obs Min = 1.57 Min = 1.49 Min = 1.53 Min = 1.37 3 3 3 3 2.5 2.5 2.5 2.5

2 2 2 2 2 2 2 2 1.5 1.5 1.5 1.5 Inflation Inflation 1.5 1.5 1.5 1.5 1 1.37 1 1.37 1 1.37 1 1.37 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 Localization Localization Localization Localization (a) EnKF (b) SEIK

Figure 5.17: Same as Fig. (5.16), but with assimilation every 2 model time steps.

Finally, to limit the impact of the sampling error, a relatively large ensemble of 200 87 members was used, supposedly enough to accurately represent the ﬁrst two moments of the state. Limitations are then assumed to be only due to model error. Since the ensemble size is larger than the state dimension, no localization is performed in this experiment, only inﬂation is applied. In this experimental setting, the EnKF- OSA estimates were found to be 10%, 12% and 13% more accurate than those of the EnKF-Reg when respectively all, half and quarter of the observations are assimilated whereas, SEIK-OSA estimates were 13%, 13% and 14% more accurate than those of SEIK-Reg for the same observational scenarios. 88 5.3 Numerical experiments with a storm surge model

This section will test the proposed SEIK-OSA scheme in a realistic setting of the ADCIRC model conﬁgured for storm surge forecasting in the Gulf of Mexico during Hurricane Ike [39].

5.3.1 An overview of ADCIRC model

Providing accurate and timely forecasts of storm surge is a problem of critical importance. Through these experiments, we consider the problem of improving the relative accuracy of short-range forecasts of storm surge using the ADvanced CIRCulation model (ADCIRC) solved on numerically coarse grids to provide timely predictions of elevated water levels. Coarse discretizations ensure a quick forecast of storm surge. However, they may lead to large numerical errors. This explains the need for assimilating data into these models. ADCIRC is a hydrodynamic circulation numerical model based on the shallow water equations that are solved using the Generalized Wave Continuity Equation (GWCE) [3]. These equations are presented in Appendix.B. The model was designed to simulate water level and current over an unstructured gridded domain. It has been under development since the 90s by the Director of the Institute of Marine Sciences, Rick Luettich, who declared that the model can be used to track the movement of the ocean, from its changes during tides to its behavior during a storm. It has been then used by organizations like FEMA (Federal Emergency Management Agency) and the U.S. Coast Guard. Few years ago, ADCIRC was used to track Hurricane Sandy and won the Department of Homeland Security’s Science and Technology Directorate’s Impact Award in 2010 and 2012. Many modiﬁcations and upgrades to the model have been latter made and ADCIRC currently enjoys very wide use among the academic community and federal agencies, such as the Army Corps of Engineers, NOAA (National Oceanic and Atmospheric Administration), and the Naval Research Laboratory. ADCIRC has an extensive 89 and successful history of storm surge prediction in coastal waters and marginal seas. Within the last decade, modeling of hurricane storm surge has become one of its principal applications [39].For more details about the model, the reader is invited to visit the model oﬃcial site (www.adcirc.org).

5.3.2 Experimental design and implementation

We conduct twin experiments in the Gulf Of Mexico using Hurricane Ike event as a study case. [3]. Hurricane Ike traveled through the Atlantic, Caribbean, and Gulf of Mexico in September of 2008 and made landfall ﬁnally along the upper Texas coast in the early morning hours of 13 September as shown in Figure (5.18). It was classiﬁed as a category-4 hurricane on 4 September 2008, and as a category-2 hurricane when it made landfall near Galveston, Texas, at 07:10 UTC 13 September 2008 [3].

Figure 5.18: Track of Hurricane Ike through the Gulf of Mexico. The circles with annotations are the locations of landfall on Sept. 13, 2008 at 07:10 UTC and the locations of the hurricane approximately 48 and 72 h before landfall. The x axis represents the longitude and the y axis the latitude in degrees.

A ﬁrst conﬁguration, is referred to as the hindcast study is implemented to generate the true states from which synthetic observations are extracted and then perturbed with random noises. 90 Synthetic data generation (Hindcast setting)

This configuration uses a high resolution grid of the study domain covering the Gulf of Mexico and the western North Atlantic sea board. It uses also high fidelity wind fields that were computed from wind data collected during the actual hurricane. This hindcast simulation is also forced with data-assimilated atmospheric pressure fields provided by the Ocean Weather, Inc (OWI). ADCIRC model was integrated using a time step of one second on a grid of 3322439 nodes corresponding to 6615381 elements defining the spatial discretization of the Gulf of Mexico and the western North Atlantic sea board. The averaged mesh element size is 1.34 Km2. Measurement data of water levels were extracted and stored every two hours, to be latter used for assimilation. These models require highly resolved finite element meshes and typically require small time steps. Consequently, running them in forecast mode will, for sure, require significant computational resources just for one forecast. So, running a large ensemble of forecasts in order to improve accuracy, would be beyond the capacities of current computational resources.

Design of the assimilation experiments

The ADCIRC configuration in these simulations which correspond to a realistic situation of the forecast model with errors , and thus, is different from the hindcast study setting. The forecast model is integrated using a coarser resolution grid including only the Gulf of Mexico and is forced with coarse global wind fields using the best possible hurricane track data obtained from the NOAA archive: ftp: //ftp.tpc.ncep.noaa.gov/atcf/archive/. The experiments are performed using a time step of ten seconds with a coarser grid of only 8006 nodes corresponding to 14269 elements, with an averaged size of 98 Km2, covering the Gulf of Mexico, as shown in Figure 5.19. 91

Figure 5.19: Finite element mesh of the Gulf of Mexico. The x axis represents the longitude and the y axis the latitude in degrees.

The results of the coarse model are compared to the solution of the hindcast study to evaluate and compare the performances of the tested ensemble ﬁlters. For the coarse model, after one day spin-up from Sept. 9, 2008 at 00:00 UTC until Sept. 10, 2008 00:00 UTC, data are assimilated every six hours until Sept. 14, 2008 at 06:00 UTC, one day after Hurricane Ike made landfall, resulting in 17 assimilation steps. Data used for assimilation were extracted using the hindcast simulation from 43 observation stations. These correspond to actual measurement sites placed by several federal agencies, including the U.S. Geological Survey (USGS) and NOAA in addition to some university research groups. Their data could be exploited in any real-time extreme event scenario. Locations of these 43 observation stations over the domain are shown in Figure 5.20. The stations are all located near shore, as models tend to underpredict the surge in coastal areas. The assimilated observations are not perfect. Their error covariance matrix is assumed diagonal ( as they are assumed to be uncorrelated), with an homogeneous standard deviation set to three centimeters. The outputs of the model are the water elevation and the two components of velocity at each node. That is, the state vector contains the elevation at each node followed by the two components of velocity. The initial ensemble is randomly selected from a long run of a model run forced only with tides. 92

Figure 5.20: Distribution of the 43 observation stations. The x axis represents the longitude and the y axis the latitude in degrees.

Hybrid SEIK and hybrid SEIK-OSA algorithms

We implemented SEIK and SEIK-OSA with ADCIRC based on a hybrid formulation of their covariances following [40]. The goal was to develop an efficient ensemble assimilation scheme for storm surge forecasting . Hybrid was already shown particularly performant in many studies even when the number of flow-dependent ensemble members is small. This is important for timely forecasting of the surge. OSA smoothing based filtering is expected to be particularly useful in such a setting where model errors and nonlinearities are expected to be consequent during the surge, the observations are limited and the flow-dependent ensemble is small.

The standard hybrid SEIK algorithm (Hybrid SEIK-Reg)

The key idea of the hybrid scheme is to express the forecast error covariance as a linear combination of a time invariant covariance and a ﬂow-dependent covariance:

f f Pen = (1 − α)Pn + αB, (5.4)

where α is a real number between 0 and 1 and B refers to a given static background covariance matrix often estimated from a climatological ensemble. Recall the expres- 93 sion of the background forecast covariance in the SEIK ﬁlter

f f f T Pn = LnGLn . (5.5)

the hybrid covariance can then be expressed as

f f Pen = (1 − α)Pn + αB

f f T T = (1 − α)LnGLn + αSbSb (5.6)

where S is a square root of B of dimension N × n with n is the number of b  x b b G 0n ×n columns of S . Now, if we deﬁne G =  b b  and Lf such that b e   en 0Ne−1×Ne−1 Inb×nb √ f f √ Len = [ 1 − αLn, αSb] , we obtain

f f f T Pen = LenGe(Len) . (5.7)

In the hybrid SEIK, the mean forecast state is updated using the same equations as

f in the standard SEIK, but using the augmented matrices Lfn and Ge

T a f f a f −1 f xn = xn + LenUfn (HnLen) Rn (yn − Hnxn), (5.8)

with −1 T a −1 f −1 f (Ufn ) = Ge + (HnLen) Rn (HnLen). (5.9)

The resampling step is only carried on the ﬂow-dependent component and is thus performed using the following equation:

a,i a p f a −1T T xn = xn + Ne − 1Ln(Cn) Ωn(:,i). (5.10) 94

where (:, i) stands for the ith column of the considered matrix and Ωn is the Ne ×Ne −1 matrix with orthonormal columns and zero column sums.

a −1 −1 f T −1 f (Un ) = G + (HnLn) Rn (HnLn), (5.11)

and

a −1 a aT (Un ) = CnCn . (5.12)

The hybrid SEIK algorithm with OSA smoothing (Hybrid SEIK-OSA)

The derivation of the hybrid SEIK-OSA follows that of the hybrid-SEIK. We therefore take

f2 f2 Pen = (1 − α)Pn + αB,

T f2 f2 T = (1 − α)Ln GLn + αSbSb . (5.13)

√ √ f2 f2 Deﬁning Len = [ 1 − αLn , αSb], we obtain

T f2 f2 f2 Pen = Len Ge(Len ) . (5.14)

The mean forecast state is then updated according to

T a f2 f2 a f2 −1 f2 xn = xn + Len Ufn (HnLen ) Rn (yn − Hnxn ), (5.15) with −1 T a −1 f2 −1 f2 (Ufn ) = Ge + (HnLen ) Rn (HnLen ), (5.16) 95 and the resampling step is performed using the following equation

T a,i a p f2 a −1 T xn = xn + Ne − 1Ln (Cn) Ωn(:,i). (5.17)

Similarly, the analysis covariance in the smoothing step is expressed as a linear combination of a time invariant covariance and a ﬂow-dependent covariance

a a Pen−1 = (1 − α)Pn−1 + αB (5.18)

√ a a √ Taking Len−1 = [ 1 − αLn−1, αSb] , we obtain

a a a T Pen−1 = Len−1Ge(Len−1) . (5.19)

The smoothing equations can be then written as

T s a a s f1 −1 f1 xn−1 = xn−1 + Len−1Uen−1(HnLen ) Rn (yn − Hnxn ), (5.20)

where −1 T s −1 f1 −1 f1 (Uen−1) = Ge + (HnLen ) Rn (HnLen ). (5.21)

The resampling step is also performed using:

s,i a p a s −1T T xn−1 = xn−1 + Ne − 1Ln−1(Cn−1) Ωn(:,i), (5.22)

with

s −1 −1 a T −1 a (Un−1) = G + (HnLn−1) Rn (HnLn−1), (5.23)

and

s −1 s s T (Un−1) = Cn−1Cn−1 . (5.24) 96 5.3.3 Results and discussion

In a storm surge problem, a particular quantity of interest is the prediction of water levels along coastal regions and during the water surge. This is why the filters are evaluated and compared based on the forecast RMSE during the surge period and along the coastal domain only. The considered RMSE is then defined in the same way as in Equation (5.2), but operating only on the coastal area and averaged over the water surge period. The coastal area is defined by the nodes that have polar coordinates such that:

−95.25 < x < −94.4 and 29 < y < 29.85

The deterministic EnKFs were shown to be more efficient than the stochastic EnKF in storm surge forecasting with limited ensemble sizes [39]. Here, we study the behavior of the hybrid SEIK filter with OSA smoothing in comparison with the standard hybrid SEIK using only half of the members,so that both filters are compared under roughly the same computational cost. Results of these experiments are reported in Figure (5.21). As one can see, the hybrid SEIK filter with OSA smoothing, implemented

Hybrid SEIK-Reg Hybrid SEIK-OSA Min = 0.96 Min = 0.88 1.2 1.2 0.3 0.3 1.15 1.15 0.25 0.25 1.1 1.1

02 1.05 02 1.05

0.15 1 0.15 1

01 0.95 01 0.95

0.9 inflation factor inflation factor 0.9 005 005 0.85 0.85 0 0 0.8 0.8 25 50 100 200 500 1000 2000 25 50 100 200 500 1000 2000 LA radius LA radius (a) Hybrid SEIK-Reg (b) Hybrid SEIK-OSA

Figure 5.21: Coastal-averaged analysis RMSE with local analysis for diﬀerent radii and inﬂation factors. The results are presented for an ensemble size of 10 with the hybrid SEIK and 5 members with the hybrid EIK-OSA for Hurricane Ike simulations. Minimum values are indicated in the title. 97

4 3 3

2.5 2.5 3 2 2

2 1.5 1.5

1 1 1 0.5 0.5 Water elevation 0 0 0 5 10 15 5 10 15 5 10 15 true state Hybrid SEIK-Reg forecast 4 4 4 Hybrid SEIK-OSA forecast

3 3 3

2 2 2

1 1 1 Water elevation 0 0 0 5 10 15 5 10 15 5 10 15 Assimilation steps Assimilation steps Assimilation steps

Figure 5.22: Plots of true and forecast states associated with the minimum of forecast RMSE at some stations located close to the landfall areas.

with only 5 members, suggests better results than the standard hybrid SEIK ﬁlter, implemented with 10 members. The minimum coastal forecast RMSE achieved by hybrid SEIK-OSA with 5 members is 0.88 compared to 0.96 with the hybrid SEIK-Reg with 10 members, which corresponds to an improvement of roughly 9% introduced by the OSA-based algorithm. Figure (5.22) shows the true state (water level) from the hindcast study at some stations located close to the landfall areas as well as the forecast states produced by the standard hybrid SEIK ﬁlter implemented with 10 members and from the hybrid SEIK-OSA with 5 members. We observe that the forecast errors increase mainly before and during the surge. The hybrid SEIK-OSA has the advantage of bringing the state closer to the truth during the surge period. To further assess the temporal evolution of the coastal RMSE during the hurricane simulation, we present in Figure (5.23), the time series of the RMSE over the coastal area. Overall, the coastal RMSE is highest during the surge and the hybrid SEIK-OSA 98

1.2 Analysis SEIK-OSA Forecast SEIK-OSA 1.1 Analysis SEIK-Reg Forecast SEIK-Reg 1

0.9

0.8

rmse 0.7

0.6

0.5

0.4

0.3 10 11 12 13 14 15 16 17 Assimilation steps

Figure 5.23: Time series of the coastal RMSE. outperforms the hybrid SEIK-Reg during this period using comparable computational cost. Indeed, with the hybrid SEIK-Reg, the maximum RMSE value is around 1.2 while with the hybrid SEIK-OSA, it decreases to 1.1. The analysis RMSEs also exhibit a similar behavior. Finally, we analyze the spatial distribution of the absolute forecast and analysis errors of free surface elevations at 06:00 UTC, 13 September 2008 (an hour before Ike made landfall at 07:10 UTC) provided by the different filters. The results are presented using the empirically determined best values of inflation factor and localization radii. In general, we notice from Figure (5.24) that the forecasts and analyses errors are large during the surge period (the model usually underpredicts the level of the surge as shown in [39]). This is expected given the coarse discretization in the forecast model that causes the dissipation of water levels and the coarse wind forcing. The hybrid SEIK-OSA improves the quality of the state estimates and brings the model closer to the true state. This might be explained by the fact that the smoothing step during the surge helps improving the background by exploiting the future data enhancing the accuracy of the analysis and consequently next forecast. 99

Figure 5.24: Plots of free surface absolute elevation errors one hour before the landfall using the best empirically determined best choices of inﬂation factor and localization.

5.4 Summary and conclusions

Performances of the standard and OSA-based EnKF and SEIK filters were tested and compared using two different numerical models, the Lorenz-96 model and a realistic storm surge forecasting model (ADCIRC) First, with L96, it was found that the performances of the SEIK-OSAQˆ (with approximate model error covariance) were severely limited by the considered approximation. This filter was not introducing enough improvement compared to the standard SEIK. The SEIK-OSA filter, however, exhibits a better behavior than SEIK-Reg in many challenging situations, including those where the ensemble size is limited, the observations are not frequent in space and time, as well as twhen the measurement error is large. With small ensembles, we found that the deterministic SEIK-OSA outperforms the stochastic OSA-based filter. When comparing the relative improvements in accuracy with respect to the standard schemes, the SEIK-OSA was also found to be more efficient than the EnKF-OSA mainly when the ensemble size is small. With ADCIRC, the hybrid SEIK-OSA was implemented and compared with hybrid SEIK-Reg in the context of a storm surge 100 prediction problem, using the Hurricane Ike event as a study case. Results of the OSA-based scheme are encouraging and led to more accurate forecast and analysis state estimates, mainly in the coastal domain and during the water surge. 101

Chapter 6

Concluding Remarks

This thesis investigated the relevance of a new ensemble filtering approach that smartly exploits the data to enhance the performances of ensemble Kalman filters. We followed the one-step-ahead smoothing (OSA) formulation of the filtering problem which involves two correction steps using the current observation at each assimilation cycle. Once the forecast is computed, the upcoming observation is first used to smooth the previous analysis state ensemble before reintegrating the model forward in time and updating the resulting members with the same observation. Computationally, the resulting ensemble filter with OSA smoothing is roughly twice more expensive than a standard one. In the linear Gaussian case, the two schemes are equivalent in the sense that they provide the same state estimates. However, in the nonlinear case, this framework is expected to enhance the filter performance as the smoothing step of the state ensemble improves the statistics of the background state and thus the forecast at the next analysis step. This framework should be particularly beneficial in situations when the ensemble is small and the model errors are large, as in a storm surge forecasting problem. The stochastic ensemble formulation of the OSA-based filtering strategy was recently proposed in [1]. In this work, two new deterministic OSA-based Singular Evolutive Interpolated EnKF (SEIK) filters are proposed. The motivation behind deriving a deterministic OSA-based filter is that this type of filters is known to be more rele- vant than the stochastic EnKF when implemented with small ensembles compared to the number of assimilated observations, a common case in oceanic and atmospheric 102 applications. The derivation of a deterministic variant of the OSA-based filtering algorithm was not straightforward since the analysis equations are functions of the model error covariance matrix, which is very poorly known in real applications. Two solutions were proposed resulting in two different OSA-based SEIK filters. First, a simple attempt to overcome this problem consisted in parameterizing the model error covariance matrix as a fraction of the forecast error covariance matrix estimated from the ensemble members. Experimentally, this approximation was not efficient in the tested experiments and the scheme did not result in significant improved estimates compared to the standard SEIK. The formulation of the second OSA-based SEIK filter (SEIK-OSA) was inspired from the stochastic EnKF-OSA algorithm. The idea was to rewrite the analysis equations that depend on the model error covariance matrix in another form that does not implicitly involve this matrix. This required neglecting a cross-covariance term in order to obtain a form of the analysis equations that is very similar to the standard deterministic ensemble Kalman filters.

Tested with the Lorenz-96 model through a number of numerical experiments to evaluate its accuracy and robustness, the SEIK-OSA algorithm was shown to outperform the standard SEIK and the stochastic EnKF-OSA. The results also clearly demonstrated that the proposed SEIK-OSA scheme is more eﬃcient and robust under diﬀerent experimental settings and scenario. This was particularly true under the challenging situations where the spatial and temporal data coverage is low, the signal to noise ratio is high and the model error is poorly known.

The hybrid SEIK-OSA was also implemented and compared against the standard hybrid SEIK for data assimilation with the ADCIRC model in the context of a storm surge prediction problem using the Hurricane Ike event as a study case. Results were shown to be encouraging and led to better forecasts and analysis mainly during the 103 surge.

The proposed SEIK-OSA scheme is easy to implement and requires only few modifica- tions to an existing SEIK code. Although, it is computationally twice more expensive, the proposed scheme leads to more accurate estimates, with the same computational cost; that is using only half the number of ensemble members of a standard algorithm. In this study, SEIK-OSA was proven efficient and robust under different experimental settings and scenarios. It should be therefore of a great benefit given its demonstrated robustness and high accuracy under challenging modeling conditions.

Potential future work will involve testing the proposed scheme with other realistic large-scale ocean data assimilation and forecasting problems which would require including various hyper-parameters as part of the estimation problem. Another future direction worth pursuing concerns extending the OSA-based formulation to coupled data assimilation problems as a generalization of the state-parameter estimation problem. 104

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APPENDICES

A Localization techniques

Local Analysis

Local analysis the update of each node j is computed locally based on the local observations within the speciﬁed radius. A local observation matrix Hσ, a local obser-

σ vation vector yn and a local observational error covariance matrix Rσ are constructed so that they take into account only the local observations for the considered node j. The Kalman gain is computed using (2.20) and replacing the global H and R by the local Hσ and Rσ as:

σ f T f T −1 Kn = Pn Hσ (HσPn Hσ + Rσ) (A.1)

The update equation for the ith ensemble at the speciﬁc node j is performed according to:

a,i f,i σ h σ,i σ f,i i xn (j) = xn (j) + Kn,(j,:) (yn − H xn ) (A.2)

where the subscript j represents the jth element of the state vector and (j; :) stands

σ,i for the jth line of the considered matrix (j = 1, 2, ..., Nx). yn is the perturbed local observation vector corresponding to the ensemble i. 109 Covariance localization when performing CL, the Kalman gain expression becomes

f T f T −1 Kn = C1 ◦ Pn Hn (C2 ◦ HnPn Hn + Rn) . (A.3)

where the elements of the correlation matrices C1 and C2 are usually calculated using a ﬁfth-order function of Gaspari and Cohn (1999) given by:

  − 1 ( b )5 + 1 ( b )4 + 5 ( b )3 − 5 ( b )2 , 0 b a  4 a 2 a 8 a 3 a 6 6  1 b 5 1 b 4 5 b 3 5 b 2 b 2 b −1 ρ = ( ) − ( ) + ( ) + ( ) − 5( ) + 4 − ( ) , a < b 6 2a  12 a 2 a 8 a 3 a a 3 a   0 , a > 2b

When calculating C1, b = kDi,jk is given by the distance between the node i and the observation station j. C1 will then be a Nx × No matrix. However, when calculating

C2, b = kDi,jk will be the distance between the observation station i and the observation station j resulting in a C2 matrix of dimension No × No. The variable a is the

q 10 length scale and is equal to 3 lc where lc is a cutoﬀ length scale chosen by the user. 110

B Shallow water equations in ADCIRC

Shallow water equations are used to model the hydrodynamic behavior of oceans and coastal areas. The two-dimensional character of a free surface flow is usually enforced by a horizontal length scale much larger than the vertical one and by a velocity field quasi-homogeneous over the water depth. Under these conditions, the 3D Reynolds averaged Navier-Stokes equations can be simplified in order to obtain the depth averaged shallow water equations. Thus, shallow water equations are the simplest form of the equations of motion that can be used to describe the horizontal structure of fluids system in ocean and atmosphere. They describe the evolution of an incompressible fluid in response to gravitational and rotational accelerations. Shallow water equations are a set of partial differential equations that describe fluid flow. They are derived from the physical conservation laws for the mass and momentum and are valid for the problems in which vertical dynamics can be neglected compared to horizontal effects. 2D shallow water model is the model which is derived from 3D model by applying depth averaging. The application of shallow water model can be found in Tsunamis prediction, atmospheric flows, storm surges and planetary flow etc. Using the Reynolds averaged Navier-Stokes equations, the hydrostatic and Boussinesq approximations with the assumption of largeness of the earth’s radius compared to the depth of the ocean and averaging over the water depth, the two- dimensional shallow water equations in spherical coordinates (λ,φ), as described in [3] can be written as:

• Continuity: ∂ζ 1 ∂ (UH) ∂ (VH cos φ) + + = 0 (B.1) ∂t R cos φ ∂λ ∂φ 111 • and horizanl momentum

dU 1 ∂[g(ζ − αη) + ps/ρ0] τSλ = fV − + − τbf U + mλ (B.2) dt R cos φ ∂λ ρ0H

dV 1 ∂[g(ζ − αη) + ps/ρ0] τSφ = −fU − + − τbf V + mφ (B.3) dt R ∂φ ρ0H

Where,

Variable Deﬁnition

t Time λ,φ Degrees longitude, latitude ζ Free-surface elevation relative to the geoid U,V Depth-averaged horizontal velocity components R Mean radius of the earth (6.3782064 × 106m) H ζ + h = water depth h Bathymetric depth relative to the geoid g Gravitational acceleration f Coriolis coeﬃcient

ps Atmospheric pressure at the free surface η Newtonian equilibrium tide potential α Eﬀective earth elasticity factor

ρ0 Reference density of water

τSλ,τSφ Applied free-surface stress

2 2 1/2 τbf Cf (U + V ) /H = bottom friction

Cf Non linear bottom friction coeﬃcient

d ∂ U ∂ V ∂ dt ∂t + R cos φ + ∂λ + R + ∂φ

νT ∂ ∂UH ∂ ∂UH mλ H [ ∂λ ( ∂λ ) + ∂φ ( ∂φ )] 112

νT ∂ ∂V H ∂ ∂V H mφ H [ ∂λ ( ∂λ ) + ∂φ ( ∂φ )]

νT Depth-averaged horizontal eddy viscosity

These equations are solved on a 2D spatial domain ω and time interval [t0,T ], with appropriate boundary and initial conditions. ADCIRC utilizes the Generalized Wave Continuity Equation (GWCE) which replaces the continuity equation (B.1) by a second order wave equation for the water elevation (in order to avoid the spurious oscillations). The equations are discretized in space using the finite element method and in time using the finite difference method. The elevation is obtained from the solution of the depth-integrated continuity equation in Generalized Wave Continuity Equation form and velocity is obtained from the solution of the momentum equations.