Ocean spectral data assimilation without background error covariance matrix

Abstract

Predetermination of background error covariance matrix B is challenging in existing ocean data assimilation schemes such as the optimal interpolation (OI). An optimal spectral decomposition (OSD) has been developed to overcome such difficulty without using the B matrix. The basis functions are eigenvectors of the horizontal Laplacian operator, pre-calculated on the base of ocean topography, and independent on any observational data and background fields. Minimization of analysis error variance is achieved by optimal selection of the spectral coefficients. Optimal mode truncation is dependent on the observational data and observational error variance and determined using the steep-descending method. Analytical 2D fields of large and small mesoscale eddies with white Gaussian noises inside a domain with four rigid and curved boundaries are used to demonstrate the capability of the OSD method. The overall error reduction using the OSD is evident in comparison to the OI scheme. Synoptic monthly gridded world ocean temperature, salinity, and absolute geostrophic velocity datasets produced with the OSD method and quality controlled by the NOAA National Centers for Environmental Information (NCEI) are also presented.

Appendices

Appendix A. Determination of H-matrix using all grid points

IDW interpolation, using all grid points, is one of the most commonly used techniques for interpolation based on the assumption that the value of h _mn in H-matrix are influenced more by the nearby points and less by the more distant points. Let

$$ {d}_n^m=\sqrt{{\left({x}^{(m)}-{x}_i\right)}^2+{\left({y}^{(m)}-{y}_j\right)}^2} $$

(41)

be the distance between the grid point (x _i , y _j ) and observational point (x ^(m), y ^(m)). The influence of the grid point x _n on the observational point x ^(m) is given by (Spepard 1968)

$$ {h}_{mn}={\left({d}_n^m\right)}^{-q}/{\displaystyle \sum_{n=1}^N{\left({d}_n^m\right)}^{-q}} $$

(42)

where q is an arbitrary positive real number called the power parameter (typically, q = 2). Another form of h _mn is given by (Franke and Nielson 1991)

$$ {h}_{mn}=\frac{{\left[\left({D}^{(m)}-{d}_n^m\right)/{D}^{(m)}{d}_n^m\right]}^2}{{\displaystyle \sum_{n=1}^N{\left[\left({D}^{(m)}-{d}_n^m\right)/{D}^{(m)}{d}_n^m\right]}^2}}, $$

(43)

where D ^(m) is the distance from the observational point x ^(m) to the most distant grid point. Equation (43) has been found to give better results than (42). As a result, c _b(x ^(m), t), is somewhat symmetric about each grid point.

Appendix B. Determination of H-matrix using neighboring grid points

Consider the position vector x = (x, y) located inside the grid cell (Fig. 14),

Fig. 14

Interpolation at an observational point r ^(m) from four neighboring grid points

x _i  ≤ x < x _i + 1,y _j  ≤ y < y _j + 1..

Mathematically, the variable c _b at r (inside the grid cell) can be represented approximately by a polynomial,

$$ {c}_b\left(\mathbf{r}\right)={\displaystyle \sum_{\alpha =0}^L{\displaystyle \sum_{\beta =0}^L{A}_{\alpha \beta }{\left(x-{x}_i\right)}^{\alpha }{\left(y-{y}_j\right)}^{\beta }}} $$

(44)

where L = 1 refers to the bilinear interpolation, and L = 3 leads to the bicubic interpolation. For the bilinear interpolation, Eq. (44) becomes

$$ {c}_b\left(\mathbf{r}\right)={A}_{00}+{A}_{10}\left(x-{x}_i\right)+{A}_{01}\left(y-{y}_j\right)+{A}_{11}\left(x-{x}_i\right)\left(y-{y}_j\right) $$

(45)

or in matrix notation,

$$ {c}_b\left(\mathbf{r}\right)=\left[1\kern0.5em \left(x-{x}_i\right)\right]\left[\begin{array}{cc}\hfill {A}_{00}\hfill & \hfill {A}_{01}\hfill \\ {}\hfill {A}_{10}\hfill & \hfill {A}_{11}\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill \left(y-{y}_j\right)\hfill \end{array}\right]. $$

(46)

Since c _b at four neighboring grid points: c _b (x _i , y _j ), c _b (x _i+1, y _j ), c _b (x _i+1, y _j ), c _b (x _i+1, y _j+1) are given, substitution of the four values into (45) leads to the determination of the four coefficients A ₀₀, A ₁₀, A ₀₁, A ₁₁. Using these coefficients, the bilinear interpolation (45) becomes

$$ \begin{array}{l}{c}_b\left(\mathbf{r}\right)=\frac{c_b\left({x}_{i+1},{y}_{j+1}\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)}\left(x-{x}_i\right)\left(y-{y}_j\right)+\frac{c_b\left({x}_{i+1},{y}_j\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)}\left(x-{x}_i\right)\left({y}_{j+1}-y\right)\\ {}+\frac{c_b\left({x}_i,{y}_{j+1}\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)}\left({x}_{i+1}-x\right)\left(y-{y}_j\right)+\frac{c_b\left({x}_i,{y}_j\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)}\left({x}_{i+1}-x\right)\left({y}_{j+1}-y\right)\end{array} $$

(47)

Let the observational point r ^(m) be located in the grid cell,

$$ {x}_i\le {x}^{(m)}<{x}_{i+1},{y}_j\le {y}^{(m)}<{y}_{j+1}. $$

Evaluation of c _b at the observational point r ^(m) using (46) leads to

$$ {c}_b\left({\mathbf{r}}^{(m)}\right)={p}_{i,j}^{(m)}{c}_b\left({x}_i,{y}_i\right)+{p}_{i+1,j}^{(m)}{c}_b\left({x}_{i+1},{y}_j\right)+{p}_{i,j+1}^{(m)}{c}_b\left({x}_i,{y}_{j+1}\right)+{p}_{i+1,j+1}^{(m)}{c}_b\left({x}_{i+1},{y}_{j+1}\right) $$

(48)

where the proportional coefficients {${p}_{i,j}^{(m)},{p}_{i+1,j}^{(m)},{p}_{i,j+1}^{(m)},{p}_{i+1,j+1}^{(m)}$} are defined by

$$ \begin{array}{ll}{p}_{i,j}^{(m)}=\frac{\left({x}_{i+1}-{x}^{(m)}\right)\left({y}_{j+1}-{y}^{(m)}\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)},\hfill & {p}_{i+1,j}^{(m)}=\frac{\left({x}^{(m)}-{x}_i\right)\left({y}_{j+1}-{y}^{(m)}\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)},\hfill \\ {}{p}_{i,j+1}^{(m)}=\frac{\left({x}_{i+1}-{x}^{(m)}\right)\left({y}^{(m)}-{y}_j\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)}\hfill & {p}_{i+1,j+1}^{(m)}=\frac{\left({x}^{(m)}-{x}_i\right)\left({y}^{(m)}-{y}_j\right)}{\left({x}_{i+1}-{x}_i\right)\left({y}_{j+1}-{y}_j\right)}.\hfill \end{array} $$

(49)

It is noted that the proportionality coefficients {p _i , j(m) , p _{i + 1 , j(m)} , p _{i , j + 1(m)} , p _{i + 1 , j + 1(m)}} depend solely on the location of the observational points (r ^(m)), and

$$ {p}_{i,j}^{(m)}+{p}_{i+1,j}^{(m)}+{p}_{i,j+1}^{(m)}+{p}_{i+1,j+1}^{(m)}=1. $$

(50)

Setting L = 3 in (44) leads to the bicubic spline interpolation,

$$ \begin{array}{l}{c}_b\left(\mathbf{r}\right)={A}_{00}+{A}_{10}\left(x-{x}_i\right)+{A}_{01}\left(y-{y}_j\right)+{A}_{11}\left(x-{x}_i\right)\left(y-{y}_j\right)\hfill \\ {}+{A}_{20}{\left(x-{x}_i\right)}^2+{A}_{02}{\left(y-{y}_j\right)}^2+{A}_{30}{\left(x-{x}_i\right)}^3\hfill \\ {}+{A}_{21}{\left(x-{x}_i\right)}^2\left(y-{y}_j\right)+{A}_{12}\left(x-{x}_i\right){\left(y-{y}_j\right)}^2+{A}_{03}{\left(y-{y}_j\right)}^3\hfill \end{array} $$

(51)

or in matrix notation,

$$ {c}_b\left(\mathbf{r}\right)=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \left(x-{x}_i\right)\hfill & \hfill {\left(x-{x}_i\right)}^2\hfill & \hfill {\left(x-{x}_i\right)}^3\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill {A}_{00}\hfill & \hfill {A}_{01}\hfill & \hfill {A}_{02}\hfill & \hfill {A}_{03}\hfill \\ {}\hfill {A}_{10}\hfill & \hfill {A}_{11}\hfill & \hfill {A}_{12}\hfill & \hfill 0\hfill \\ {}\hfill {A}_{20}\hfill & \hfill {A}_{21}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {A}_{30}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill \left(y-{y}_j\right)\hfill \\ {}\hfill {\left(y-{y}_j\right)}^2\hfill \\ {}\hfill {\left(y-{y}_j\right)}^3\hfill \end{array}\right] $$

(52)

which is rewritten by

$$ \begin{array}{l}{c}_b\left(\mathbf{r}\right)=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill x\hfill & \hfill {x}^2\hfill & \hfill {x}^3\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill -{x}_i\hfill & \hfill {x}_i^2\hfill & \hfill -{x}_i^3\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill -2{x}_i\hfill & \hfill 3{x}_i^2\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -3{x}_i\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\\ {}\left[\begin{array}{cccc}\hfill {A}_{00}\hfill & \hfill {A}_{01}\hfill & \hfill {A}_{02}\hfill & \hfill {A}_{03}\hfill \\ {}\hfill {A}_{10}\hfill & \hfill {A}_{11}\hfill & \hfill {A}_{12}\hfill & \hfill 0\hfill \\ {}\hfill {A}_{20}\hfill & \hfill {A}_{21}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {A}_{30}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -{y}_j\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {y}_j^2\hfill & \hfill -2{y}_j\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill -{y}_j^3\hfill & \hfill 3{y}_j^2\hfill & \hfill 3{y}_j\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill y\hfill \\ {}\hfill {y}^2\hfill \\ {}\hfill {y}^3\hfill \end{array}\right]\end{array} $$

(53)

Determination of the 10 coefficients (A ₀₀, A ₀₁, A ₀₂, A ₀₃, A ₁₀, A ₁₁, A ₁₂, A ₂₀, A ₂₁, A ₃₀) requires not only the values,

$$ \begin{array}{l}{A}_{00}={c}_b\left({x}_i,{y}_j\right),\hfill \\ {}{A}_{00}+{A}_{10}\varDelta x+{A}_{20}{\left(\varDelta x\right)}^2+{A}_{30}{\left(\varDelta x\right)}^3={c}_b\left({x}_{i+1},{y}_j\right),\hfill \\ {}{A}_{00}+{A}_{01}\varDelta y+{A}_{02}{\left(\varDelta y\right)}^2+{A}_{03}{\left(\varDelta y\right)}^3={c}_b\left({x}_i,{y}_{j+1}\right),\hfill \\ {}\begin{array}{l}{A}_{00}+{A}_{10}\varDelta x+{A}_{01}\varDelta y+{A}_{11}\Delta \mathrm{x}\Delta \mathrm{y}+{A}_{20}{\left(\varDelta x\right)}^2+{A}_{02}{\left(\varDelta y\right)}^2\\ {}+{A}_{30}{\left(\varDelta x\right)}^3+{A}_{21}{\left(\varDelta x\right)}^2\varDelta y+{A}_{12}\varDelta x{\left(\varDelta y\right)}^2+{A}_{03}{\left(\varDelta y\right)}^3\\ {}={c}_b\left({x}_{i+1},{y}_{j+1}\right),\end{array}\hfill \end{array} $$

(54)

but also the derivatives at the neighboring grid points

$$ \begin{array}{l}{A}_{10}=\partial {c}_b\left({x}_i,{y}_j\right)/\partial x=\left[{c}_b\left({x}_{i+1},{y}_j\right)-{c}_b\left({x}_{i-1},{y}_j\right)\right]/2\varDelta x,\hfill \\ {}{A}_{01}=\partial {c}_b\left({x}_i,{y}_j\right)/\partial y=\left[{c}_b\left({x}_i,{y}_{j+1}\right)-{c}_b\left({x}_i,{y}_{j-1}\right)\right]/2\varDelta y,\hfill \\ {}\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{l}{A}_{10}+2\left(\varDelta x\right){A}_{20}+3{\left(\varDelta x\right)}^2{A}_{30}=\partial {c}_b\left({x}_{i+1},{y}_j\right)/\partial x\\ {}=\left[{c}_b\left({x}_{i+2},{y}_j\right)-{c}_b\left({x}_i,{y}_j\right)\right]/2\varDelta x,\end{array}\hfill \end{array}\hfill \\ {}\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{l}{A}_{10}+\left(\varDelta y\right){A}_{11}+{\left(\varDelta y\right)}^2{A}_{12}=\partial {c}_b\left({x}_i,{y}_{j+1}\right)/\partial x\\ {}=\left[{c}_b\left({x}_{i+1},{y}_{j+1}\right)-{c}_b\left({x}_{i-1},{y}_{j+1}\right)\right]/2\varDelta x,\end{array}\hfill \end{array}\hfill \\ {}\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{l}{A}_{01}+\left(\varDelta x\right){A}_{11}+{\left(\varDelta x\right)}^2{A}_{21}=\partial {c}_b\left({x}_{i+1},{y}_j\right)/\partial y\\ {}=\left[{c}_b\left({x}_{i+1},{y}_{j+1}\right)-{c}_b\left({x}_{i+1},{y}_{j-1}\right)\right]/2\varDelta y,\end{array}\hfill \end{array}\hfill \\ {}\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{l}{A}_{01}+2\left(\varDelta y\right){A}_{01}+3{\left(\varDelta y\right)}^2{A}_{03}=\partial {c}_b\left({x}_i,{y}_{j+1}\right)/\partial y\\ {}=\left[{c}_b\left({x}_i,{y}_{j+2}\right)-{c}_b\left({x}_i,{y}_j\right)\right]/2\varDelta y.\end{array}\hfill \end{array}\hfill \end{array} $$

(55)

The solution of the above set of 10 linear algebraic Eqs. (54) and (55) leads to the determination of the 10 coefficients (A ₀₀, A ₀₁, A ₀₂, A ₀₃, A ₁₀, A ₁₁, A ₁₂, A ₂₀, A ₂₁, A ₃₀). It is noted that values of c _b at the 10 neighboring grid points (x _i , y _j ), (x _i+1, y _j ), (x _i , y _j+1), (x _i+1, y _j+1), (x _i-1, y _j ), (x _i , y _j-1), (x _i+2, y _j ), (x _i-1, y _j+1), (x _i+1, y _j-1), (x _i , y _j+2) are used to solve (54) and (55). Following (53), interpolation of c _b at the 10 neighboring grid points on the observational r ^(m) = (x ^(m), y ^(m)) using the bicubic interpolation is given by

$$ \begin{array}{l}{c}_b\left({\mathbf{r}}^{(m)}\right)=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill {x}^{(m)}\hfill & \hfill {\left({x}^{(m)}\right)}^2\hfill & \hfill {\left({x}^{(m)}\right)}^3\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill -{x}_i\hfill & \hfill {x}_i^2\hfill & \hfill -{x}_i^3\hfill \\ {}\hfill 0\hfill & \hfill 1\hfill & \hfill -2{x}_i\hfill & \hfill 3{x}_i^2\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -3{x}_i\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\\ {}\left[\begin{array}{cccc}\hfill {A}_{00}\hfill & \hfill {A}_{01}\hfill & \hfill {A}_{02}\hfill & \hfill {A}_{03}\hfill \\ {}\hfill {A}_{10}\hfill & \hfill {A}_{11}\hfill & \hfill {A}_{12}\hfill & \hfill 0\hfill \\ {}\hfill {A}_{20}\hfill & \hfill {A}_{21}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {A}_{30}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -{y}_j\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {y}_j^2\hfill & \hfill -2{y}_j\hfill & \hfill 1\hfill & \hfill 0\hfill \\ {}\hfill -{y}_j^3\hfill & \hfill 3{y}_j^2\hfill & \hfill 3{y}_j\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill {y}^{(m)}\hfill \\ {}\hfill {\left({y}^{(m)}\right)}^2\hfill \\ {}\hfill {\left({y}^{(m)}\right)}^3\hfill \end{array}\right]\end{array} $$

(56)

Thus, an equation similar to (48) can be written for evaluating c _b at the observational point r ^(m) with the known 10 coefficients (A ₀₀, A ₀₁, A ₀₂, A ₀₃, A ₁₀, A ₁₁, A ₁₂, A ₂₀, A ₂₁, A ₃₀,

$$ \begin{array}{l}{c}_b\left({\mathbf{r}}^{(m)}\right)={p}_{i,j}^{(m)}{c}_b\left({x}_i,{y}_j\right)+{p}_{i+1,j}^{(m)}{c}_b\left({x}_{i+1},{y}_j\right)+{p}_{i,j+1}^{(m)}{c}_b\left({x}_i,{y}_{j+1}\right)\hfill \\ {}+{p}_{i+1,j+1}^{(m)}{c}_b\left({x}_{i+1},{y}_{j+1}\right)+{p}_{i-1,j}^{(m)}{c}_b\left({x}_{i-1},{y}_j\right)+{p}_{i,j-1}^{(m)}{c}_b\left({x}_i,{y}_{j-1}\right)+{p}_{i,+2j}^{(m)}{c}_b\left({x}_{i+2},{y}_j\right)\hfill \\ {}+{p}_{i-1,j+1}^{(m)}{c}_b\left({x}_{i-1},{y}_{j+1}\right)+{p}_{i+1,j-1}^{(m)}{c}_b\left({x}_{i+1},{y}_{j-1}\right)+{p}_{i,j+2}^{(m)}{c}_b\left({x}_i,{y}_{j+2}\right)\hfill \end{array} $$

(57)

where the 10 corresponding coefficients {p _i , j(m) , p _{i + 1 , j(m)} , p _{i , j + 1(m)} , p _{i + 1 , j + 1(m)}, p _{i − 1 , j(m)}, p _{i , j − 1(m)}, p _{i ,  + 2j(m)}, p _{i − 1 , j + 1(m)}, p _{i + 1 , j − 1(m)}, p _{i , j + 2(m)}} are analytically determined and depends solely on the location of the observational points (r ^(m)), and

$$ {p}_{i,j}^{(m)}+{p}_{i+1,j}^{(m)}+{p}_{i,j+1}^{(m)}+{p}_{i+1,j+1}^{(m)}+{p}_{i-1,j}^{(m)}+{p}_{i,j-1}^{(m)}+{p}_{i,+2j}^{(m)}+{p}_{i-1,j+1}^{(m)}+{p}_{i+1,j-1}^{(m)}+{p}_{i,j+2}^{(m)} $$

(58)

Since only 10 neighboring grid points are used to interpolate at the observational point r ^(m) using the bicubic interpolation, the matrix H has only 10 non-zero values in each row. However, it is too tedious to write it out.

Appendix C. Basis functions

As pointed by Chu et al. (2015), three necessary conditions should be satisfied in selection of basis functions {ϕ _k (r)} as follows: (i) satisfaction of the same homogeneous boundary condition of the assimilated variable anomaly, (ii) orthonormality, and (iii) independence on the assimilated variables. The first necessary condition requires the same boundary condition for (c − c _b ) and the basis functions {ϕ _k }. The second necessary condition is given by

$$ {\displaystyle \underset{\varGamma }{\iint }{\phi}_k\left(\mathbf{r}\right){\phi}_{k\prime }}\left(\mathbf{r}\right)d\mathbf{r}={\delta}_{kk\prime }, $$

(59)

where δ _kk′ is the Kronecker delta,

$$ {\delta}_{kk\prime }=\left\{\begin{array}{c}\hfill 0\kern1em \mathrm{if}\kern0.5em k\ne k^{\prime}\hfill \\ {}\hfill 1\kern1.25em \mathrm{if}\kern0.5em k=k^{\prime}\hfill \end{array}\right.. $$

(60)

Due to their independence on the assimilated variable (the third necessary condition), the basis functions are available prior to the data assimilation.

The basis functions are the eigenvectors {ϕ _k } of the Laplacian operator with the same boundary condition as the variable anomaly (c − c _b ),

$$ {\nabla}^2{\phi}_k=-{\lambda}_k{\phi}_{k,\kern1.5em }\left[{b}_1\left(\tau \right)\mathbf{e}\cdotp \nabla {\phi}_k+{b}_2\left(\tau \right){\phi}_k\right]\Big|{}_{\varGamma }=0,\kern0.75em k=1,\dots, \infty . $$

(61)

Here, {λ _k } are the eigenvalues, e is the unit vector normal to the boundary; τ denotes a moving point along the boundary, and [b ₁(τ) ,  b ₂(τ)] are parameters varying with τ. The boundary condition in (61) becomes the Dirichlet boundary condition when b ₁ = 0, and the Neumann boundary conditions when b ₂ = 0. As pointed by Chu et al. (2015), different variable anomalies have different [b ₁(τ) ,  b ₂(τ)]. For example, the temperature, salinity, and velocity potential anomalies have b ₂ = 0 for the rigid boundary and b ₁ = 0 for the open boundary. However, the anomaly has b ₁ = 0 for the rigid boundary and b ₂ = 0 for the open boundary. It is obvious that the eigenvectors {ϕ _k } are orthonormal and independent of the assimilated variables.

Appendix D. Vapnik-Chervonenkis dimension for mode truncation

The Vapnik-Chervonenkis dimension (Vapnik 1983; Chu et al. 2003a, 2015) is to seek the optimal mode truncation on the base of the first term of the analysis error (23),

$$ {J}_{tr}=\left\langle \left[{\boldsymbol{\upvarepsilon}}_K^T\mathbf{F}{\boldsymbol{\upvarepsilon}}_K\right]\right\rangle =\frac{{\displaystyle \sum_{n=1}^N{\left[{f}_n\left({D}_n-{D}_n^{(K)}\right)\right]}^2}}{N-1} $$

(62)

with the cost function

$$ \begin{array}{l}{J}_K={J}_{tr}+\mu \left(K,M,\alpha \right),\\ {}\mu \left(K,M,\alpha \right)={J}_{*}\sqrt{\frac{\left[ \ln \left(2M/K\right)+1\right]- \ln \left(\alpha /M\right)}{M/K}},\kern0.75em K=1,2,\dots, \infty \end{array} $$

(63)

Here, α (≪1) is the significance level. J _* is the upper bound of J _tr . For a given M, J _tr decreases monotonically with K; μ increases with K if α is given. The optimal mode truncation is through the minimization of the cost function,

$$ \underset{K}{\mathit{\min}}\left(\ {J}_K\right)={J}_{K_{\mathrm{opt}}}. $$

(64)

This method neglects observational error [only first term of (23) considered] and ignores the model resolution (represented by the total number of grid points N). The ratio of observational points (M) and the spectral truncation (K) is the key to determine the optimal mode truncation K _OPT.

Appendix E. B matrix

The B matrix is often established based on the assumption of statistical stationarity and homogeneity of the reconstructed field with a simple covariance function, for example Bretherton et al. (1976) proposed

$$ \mathbf{B}={\left[{b}_{ij}\right]}_{N\times N},\kern1em {b}_{ij}=\left(1-\frac{r_{ij}^2}{r_b^2}\right) \exp \left(-\frac{r_{ij}^2}{r_a^2}\right),\kern0.75em {r}_{ij}^2={\left|{\mathbf{r}}_i-{\mathbf{r}}_j\right|}^2,\kern0.5em {r}_b>{r}_a, $$

(65)

depending on distances only. Here, r _ij is the distance between the two grid points r _i and r _aj ; r _ay and r _b are the decorrelation scale and zero crossing. To conduct the OI data assimilation, the three parameters (e _o , r _a , r _b ) need to be defined by user. Chu et al. (1997, 2002) compute auto-correlation functions from historical observational data to fit the Gaussian function and get de-correlation scales for the B matrix. Recent studies show that some variables such as upper ocean current speed do not satisfy the normal distribution, but the Weibull distribution (Chu 2008, 2009).