julia

Basic equation in Lorenz (1996)

The basic equation in Lorenz (1996) is expressed as: \begin{equation} \dfrac{dX_k}{dt} =\left(X_{k+1}-X_{k-2} \right) X_{k-1}-X_k+F,\quad (k=1,\cdots ,N), \tag{L.1} \label{eq:exam-3-2-3-1} \end{equation} where the prediction variable \(X_k\) has the periodic condition: \begin{equation} X_{k-N}=X_k=X_{k+N}. \tag{L.2} \label{eq:exam-3-2-3-2} \end{equation}

Time integration for the forecast

The Lorenz96 equation (\ref{eq:exam-3-2-3-1}): \begin{equation} \dfrac{d\textbf{x}}{dt} =f(t,\textbf{x}),\quad \textbf{x} \equiv (X_1,\cdots ,X_k,\cdots ,X_N)^T \tag{F.1} \label{eq:app-1-1} \end{equation} is integrated by the standard 4th-order Runge-Kutta scheme: \begin{equation} \textbf{x} _{i+1} =\textbf{x} _i+\dfrac{\Delta t}{6} \left(\textbf{k} _1+2\textbf{k} _2+2\textbf{k} _3+\textbf{k} _4 \right) =M(\textbf{x} _i) , \nonumber \end{equation} \begin{equation} \textbf{k} _1\equiv f(t_i,\textbf{x} _i) , \nonumber \end{equation} \begin{equation} \textbf{k} _2\equiv f(t_i+\Delta t/2,\textbf{x} _i+(\Delta t/2)\textbf{k} _1) , \nonumber \end{equation} \begin{equation} \textbf{k} _3\equiv f(t_i+\Delta t/2,\textbf{x} _i+(\Delta t/2)\textbf{k} _2) , \nonumber \end{equation} \begin{equation} \textbf{k} _4\equiv f(t_i+\Delta t,\textbf{x} _i+\Delta t\textbf{k} _3) . \nonumber \end{equation}

Initialization and configuration of an observation system simulation experiment (OSSE)

• Spin-up experiment (required by making initial conditions for the perfect and forecast models)
  • Initial condition: \(\textbf{x}^{\mathrm{s}}_0=\left[1.1,\; 1.0,\; \cdots ,\; 1.0 \right] ^T \)
  • Time integration: \(N_s\) steps (for sufficiently long period)
• Perfect model experiment (\(\textbf{x}^{\mathrm{t}}_i\))
  • Initial condition: \(\textbf{x}^{\mathrm{t}}_0=\textbf{x}^{\mathrm{s}}_Ns \)
  • Time integration: \(N_t\) steps
• Forecast model experiment (\(\textbf{x}^{\mathrm{f}}_i\))
  • Initial condition: Temporal average of \(\textbf{x}^{\mathrm{s}} \) for a period of 0 to \(N_s\)
  • Time integration: \(N_t\) steps
• Pseudo-observation (\(\textbf{y}^{\mathrm{o}}_i\))
  • Observation grids are located at model grids (without any interpolations)
  • \(\textbf{y}^{\mathrm{o}}_i=\textbf{x}^{\mathrm{t}}_i+\textbf{r} \)
    • \(\textbf{r} \) is composed of random perturbations with a normal distribution (\(N(0, \sigma _R)\))
    • \(\sigma _R \) is standard deviations of observation

The data assimilation-forecast cycles

(Extended) Kalman Filter

• Forecast equations: \begin{equation} \textbf{x}^{\mathrm{f}} _{i+1}=M(\textbf{x}^{\mathrm{a}} _i), \tag{KF.1} \label{eq:KF-1} \end{equation} \begin{equation} \textbf{P} ^{\mathrm{f}} _{i+1}\approx \textbf{M} \textbf{P} ^{\mathrm{a}} _i \textbf{M} ^T. \tag{KF.2} \label{eq:KF-2} \end{equation}

• Kalman gain (\(\textbf{K} _i\)) equation: \begin{equation} \textbf{K} _i=\textbf{P} ^{\mathrm{f}} _i \textbf{H} ^T_i\left(\textbf{H} _i \textbf{P} ^{\mathrm{f}} _i \textbf{H} ^T_i+\textbf{R} _i \right) ^{-1}. \tag{KF.3} \label{eq:KF-3} \end{equation}

• Analysis equations: \begin{equation} \textbf{x}^{\mathrm{a}} _i=\textbf{x}^{\mathrm{f}} _i+\textbf{K} _i \left[\textbf{y}^{\mathrm{o}} _i -H_i(\textbf{x}^{\mathrm{f}} _i) \right] . \tag{KF.4} \label{eq:KF-4} \end{equation} \begin{equation} \textbf{P} ^{\mathrm{a}} _i=\left(\textbf{I} -\textbf{K} _i\textbf{H} _i \right) \textbf{P} ^{\mathrm{f}} _i. \tag{KF.5} \label{eq:KF-5} \end{equation}

• Covariance inflation: \begin{equation} \textbf{P}^{\mathrm{a}} _i\leftarrow (1+\Delta )\textbf{P}^{\mathrm{a}} _i,\quad (0<\Delta ). \tag{KF.6} \label{eq:KF-6} \end{equation}

• Symbols

  • \(\textbf{x}^{\mathrm{f}}\): Forecast (i.e., first guess) variables (\(N\)-dimension vector),
  • \(\textbf{x}^{\mathrm{a}}\): Analysis variables (\(N\)-dimension vector),
  • \(\textbf{y}^{\mathrm{o}}\): Observation variables (\(p\)-dimension vector),
  • \(\textbf{P} ^{\mathrm{f}} \): Background covariance matrix (\(N\times N\)),
  • \(\textbf{P} ^{\mathrm{a}} \): Analysis covariance matrix (\(N\times N\)),
  • \(\textbf{R} \): Observation covariance matrix (\(p\times p\)),
  • \(M\): Model operator for time integration (linear or non-linear),
  • \(\textbf{M} \equiv \partial M/\partial \textbf{x} \): Tangent linear operator corresponding to the Model operator (\(N\times N\)),
  • \(H\): Observation operator (linear or non-linear),
  • \(\textbf{H} \equiv \partial H/\partial \textbf{x} \): Tangent linear operator corresponding to the Observation operator (\(p\times N\)),

Singular Evolutive Extended Kalman (SEEK) Filter

• Forecast equations: \begin{equation} \textbf{x}^{\mathrm{f}} _{i+1}=M(\textbf{x}^{\mathrm{a}} _i), \tag{SEEKF.1} \label{eq:SEEKF-1} \end{equation} \begin{equation} \hat{\textbf{U}}’ _{i+1}\approx \textbf{M} \hat{\textbf{U}} _i. \tag{SEEKF.2} \label{eq:SEEKF-2} \end{equation}

• Kalman gain (\(\textbf{K} _i\)) equation: \begin{equation} \textbf{K} _i=\hat{\textbf{U}}’ _i\hat{\textbf{D}}’ _i(\hat{\textbf{U}}’ _i)^T \textbf{H} ^T_i\textbf{R} ^{-1}_i. \tag{SEEKF.3} \label{eq:SEEKF-3} \end{equation}

• Analysis equations: \begin{equation} \textbf{x}^{\mathrm{a}} _i=\textbf{x}^{\mathrm{f}} _i+\textbf{K} _i \left[\textbf{y}^{\mathrm{o}} _i -H_i(\textbf{x}^{\mathrm{f}} _i) \right] , \tag{SEEKF.4} \label{eq:SEEKF-4} \end{equation} \begin{equation} \hat{\textbf{D}}’ _i\approx \hat{\textbf{D}} _{i-1}-\hat{\textbf{D}} _{i-1}(\hat{\textbf{U}}’ _i)^T\textbf{H} ^T_i\left[\textbf{H} _i\hat{\textbf{U}}’ _i\hat{\textbf{D}} _{i-1}(\hat{\textbf{U}}’ _i)^T\textbf{H} ^T_i+\textbf{R} _i \right] ^{-1}\textbf{H} _i\hat{\textbf{U}}’ _i\hat{\textbf{D}} _{i-1}, \tag{SEEKF.5} \label{eq:SEEKF-5} \end{equation} \begin{equation} \hat{\textbf{D}}’ _i=\textbf{L} \textbf{L} ^T, \quad (\mathrm{Cholesky\; decomposition}), \tag{SEEKF.6} \label{eq:SEEKF-6} \end{equation} \begin{equation} \textbf{L} ^T(\hat{\textbf{U}}’ _i)^T\hat{\textbf{U}}’ _i\textbf{L} =\textbf{V} \textbf{E} \textbf{V} ^T, \quad (\mathrm{Eigenvalue\; decomposition}), \tag{SEEKF.7} \label{eq:SEEKF-7} \end{equation} \begin{equation} \hat{\textbf{U}} _i=\hat{\textbf{U}}’ _i\textbf{L} \textbf{V} \textbf{E} ^{-1/2},\quad \hat{\textbf{D}} _i=\textbf{E} . \tag{SEEKF.8} \label{eq:SEEKF-8} \end{equation}

• Symbols

  • \(\textbf{x}^{\mathrm{f}}\): Forecast (i.e., first guess) variables (\(N\)-dimension vector),
  • \(\textbf{x}^{\mathrm{a}}\): Analysis variables (\(N\)-dimension vector),
  • \(\textbf{y}^{\mathrm{o}}\): Observation variables (\(p\)-dimension vector),
  • \(\textbf{P} ^{\mathrm{f}} \): Background covariance matrix (\(N\times N\)),
  • \(\textbf{P} ^{\mathrm{a}} \): Analysis covariance matrix (\(N\times N\)),
  • \(\textbf{R} \): Observation covariance matrix (\(p\times p\)),
  • \(M\): Model operator for time integration (linear or non-linear),
  • \(\textbf{M} \equiv \partial M/\partial \textbf{x} \): Tangent linear operator corresponding to the Model operator (\(N\times N\)),
  • \(H\): Observation operator (linear or non-linear),
  • \(\textbf{H} \equiv \partial H/\partial \textbf{x} \): Tangent linear operator corresponding to the Observation operator (\(p\times N\)),

Local Ensemble Transform Kalman Filter (LETKF)

• Forecast equations (for each ensemble member, m): \begin{equation} \textbf{X}^{\mathrm{f}} _{i+1}=M(\textbf{X}^{\mathrm{a}} _i), \tag{LETKF.1} \label{eq:LETKF-1} \end{equation} \begin{equation} \textbf{X}\equiv \left[\textbf{x}^{(1)}|\cdots |\textbf{x}^{(m)} \right] =\overline{\textbf{X}} +\delta \textbf{X} , \tag{LETKF.2} \label{eq:LETKF-2} \end{equation} where \(\overline{(\; )}\) means ensemble mean.

• Analysis equations: \begin{equation} \textbf{X}^{\mathrm{a}} _i=\overline{\textbf{X}} ^{\mathrm{f}} _i+\delta \textbf{X} ^{\mathrm{f}} _i\left[\textbf{U} \textbf{D} ^{-1}\textbf{U} ^T(\textbf{H} _i\delta \textbf{X} ^{\mathrm{f}} _i)^T(\textbf{R} _i)^{-1}(\textbf{Y} ^{\mathrm{o}}_i-\overline{H_i(\textbf{X} ^{\mathrm{f}} _i)} )+\; \sqrt[]{m-1} \textbf{U} \textbf{D} ^{1/2}\textbf{U} ^T \right] , \tag{LETKF.3} \label{eq:LETKF-3} \end{equation} \begin{equation} (m-1)\textbf{I}+(\textbf{H} _i\delta \textbf{X} ^{\mathrm{f}} _i)^T(\textbf{R} _i)^{-1}\textbf{H} _i\delta \textbf{X} ^{\mathrm{f}} _i=\textbf{U} \textbf{D} \textbf{U} ^T, \qquad (\mathrm{Eigenvalue\; decomposition}). \tag{LETKF.4} \label{eq:LETKF-4} \end{equation}

• Sub equations (Not required in the analysis procedure): \begin{equation} \textbf{K} _i=\delta \textbf{X} ^{\mathrm{f}} _i\textbf{U} \textbf{D} ^{-1}\textbf{U} ^T(\textbf{H} _i\delta \textbf{X} ^{\mathrm{f}} _i)^T(\textbf{R} _i)^{-1} . \tag{LETKF.5} \label{eq:LETKF-5} \end{equation} \begin{equation} \textbf{P} \equiv \dfrac{1}{m-1} (\delta \textbf{X} )(\delta \textbf{X} )^T. \tag{LETKF.6} \label{eq:LETKF-6} \end{equation}

• Covariance inflation (Multiplicative inflation): \begin{equation} \delta \textbf{X}^{\mathrm{a}} _i\leftarrow (1+\Delta )\delta \textbf{X}^{\mathrm{a}} _i,\quad (0<\Delta ). \tag{LETKF.7} \label{eq:LETKF-7} \end{equation}

• Symbols

  • \(\textbf{x}^{\mathrm{f}(k)}\): Forecast (i.e., first guess) variables in the \(k\)-th ensemble member (\(N\)-dimension vector),
  • \(\textbf{x}^{\mathrm{a}(k)}\): Analysis variables in the \(k\)-th ensemble member (\(N\)-dimension vector),
  • \(\textbf{Y}^{\mathrm{o}}\): Observation variables (\(p\times m\)-dimension vector),
  • \(\textbf{R} \): Observation covariance matrix (\(p\times p\)),
  • \(M\): Model operator for time integration (linear or non-linear),
  • \(H\): Observation operator (linear or non-linear),
  • \(\textbf{H} \equiv \partial H/\partial \textbf{x} \): Tangent linear operator corresponding to the Observation operator (\(p\times N\)),
  • \(m\): Total ensemble member.

A hybrid Ensemble Kalman Filter (EnKF)

(Under construction)

• Background covariance \begin{equation} \textbf{P} ^{\mathrm{f}} _i=\beta \textbf{P} _{\mathrm{stat}}+(1-\beta )\textbf{P} ^{\mathrm{f}} _{\mathrm{flow},i}, \quad (0\leq \beta \leq 1). \tag{HEnKF.1} \label{eq:HEnKF-1} \end{equation}

• Symbols

  • \(\textbf{x}^{\mathrm{f}(k)}\): Forecast (i.e., first guess) variables in the \(k\)-th ensemble member (\(N\)-dimension vector),
  • \(\textbf{x}^{\mathrm{a}(k)}\): Analysis variables in the \(k\)-th ensemble member (\(N\)-dimension vector),
  • \(\textbf{Y}^{\mathrm{o}}\): Observation variables (\(p\times m\)-dimension vector),
  • \(\textbf{P}_{\mathrm{stat}} \): Statistical or climatorogical background covariance matrix (\(N\times N\)),
  • \(\textbf{P} ^{\mathrm{f}} _{\mathrm{flow},i} \): Flow-dependent background covariance matrix (\(N\times N\)),
  • \(\textbf{R} \): Observation covariance matrix (\(p\times p\)),
  • \(M\): Model operator for time integration (linear or non-linear),
  • \(H\): Observation operator (linear or non-linear),
  • \(\textbf{H} \equiv \partial H/\partial \textbf{x} \): Tangent linear operator corresponding to the Observation operator (\(p\times N\)),
  • \(m\): Total ensemble member.

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