Ocean spectral data assimilation without background error covariance matrix

Transcription

1 Source File Click here to download Manuscript osd-lieg-od--r.docx Click here to view linked References Ocean spectral data assimilation without background error covariance matrix Peter C. Chu*, Chenwu Fan, and Tetyana Margolina Naval Ocean Analysis and Prediction Laboratory, Department of Oceanography Naval Postgraduate School, Monterey, California, USA *Corresponding author address: Department of Oceanography, Naval Postgraduate School, Monterey, CA. pcchu@nps.edu 1

2 Abstract Predetermination of background error covariance matrix B is challenging in existing ocean data assimilation schemes such as the optimal interpolation (). An optimal spectral decomposition () 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 D fields of large and small mesoscale eddies with white Gaussian noises inside a domain with rigid and curved boundaries are used to demonstrate the capability of the method. The overall error reduction using the is evident in comparison to the scheme. Synoptic monthly gridded world ocean temperature, salinity, and absolute geostrophic velocity datasets produced with the method and quality controlled by the NOAA National Centers for Environmental Information (NCEI) are also presented.

3 Introduction In ocean data assimilation (or analysis), the coordinates (x, y, z) are usually represented by the position vector r with grid points represented by rn, n = 1,,, N, and observational locations represented by r (m), m = 1,, M. Here, N is the total number of the grid points, and M is the total number of observational points. A single or multiple variables c = (u, v, T, S, ), no matter two or three dimensional, can be ordered by grid point and by variable, forming a single vector of length NP with N the total number of grid points and P the number of variables. For multiple variables, non-dimensionalization is conducted before forming a single vector c (Chu et al. 0) with true, analysis, and background fields (ct, ca, cb) and observational data (co) being represented by N and M dimensional vectors, (1) () ( ) c T,,,,( 1 ),,,( ),...,,,( ), T ( ), ( ),..., ( M tabctab r ctab r ctab rn c o co r co r co r ), (1) where the superscript T means transpose. The innovation (or called the observational increment Minimization of the analysis error variance d c Hc, () o represents the difference between the observational and background data at the observational points r (m). Here, H =[hmn] is an M N linear observation operator matrix converting the background field cb (at the grid points, rn) into first guess observations at the observational points r (m) (Fig. 1). The analysis error (εa) and observational error (εo) are defined by b T ε c c, ε H c c, (a) a a t o o t which are evaluated at the grid points. The two errors are usually independent of each other, N T 1 εε o a 0, N 1. (b) n1

4 E gives the optimal analysis field ca for the true field ct. T εε min () a a A common practice in ocean data assimilation (or analysis) is to use a N M weight matrix W= [wnm] to blend cb (at the grid points rn) with innovation d (at observational points r (m) ) (Evensen, 00; Tang and Kleeman 00; Chu et al. 00a, 0; Galanis et al. 00; Oke et al. 00; Han et al. 01; Yan et al. 0) c c Wd. () a Minimization of the analysis error variance with respect to weights, determines the weight matrix b E / w nm 0. () 1 WBH T ( HBH T R ). () Here, B is the N N background error covariance matrix; R is the M M observational error covariance matrix and is usually simplified as a product of an observational error variance ( e o ) and an identity matrix I, R e o I. () Substitution of () into () leads to the optimal interpolation () equation, T T 1 ca cb BH ( HBH R) d, () which produces the analysis field ca from the innovation d. The challenge for the method is the determination of the background error covariance matrix B. An alternative approach is to use a spectral method with lateral boundary (Г) information to decompose the variable anomaly at the grid points [c(rn) - cb(rn)] into (Chu et al. 0),

5 K c ( r ) c ( r ) s ( r ), s ( r ) a r, () a n b n K n K n k k n k 1 where {ϕk} are basis functions; K is the mode truncation. The eigenvectors of the Laplace operator with the same lateral boundary condition of (c cb) can be used as the set of the basis functions {ϕk} and written in matrix (Chu et al. 0) Φ 1( r1) ( r1)... K ( r1) ( ) ( )... ( ) r r r. () ( rn) ( rn)... K( rn) 1 K kn For a given mode truncation K, minimization of the analysis error variance () with respect to the spectral coefficients E / a 0, k 1,..., K (1) K gives the spectral ocean data assimilation equation (Chu et al. 0), k T T 1 T a b c c FΦ ΦFΦ ΦH d, (1) where F is an N N (diagonal) observational contribution matrix f f F, fn fn f n M m1 h nm. () Here, the matrices Φ, F, and H are all given in comparison to the equation () where the background error covariance matrix B needs to be determined. This spectral method has been proven effective for the ocean data analysis. Chu el al. (00a, b) named the spectral method as the optimal spectral decomposition (). With it, several new ocean phenomena have been identified from observational data such as a bi-modal

6 structure of chlorophyll-a with winter/spring (February March) and fall (September October) blooms in the Black Sea (Chu et al. 00a), fall winter recurrence of current reversal from westward to eastward on the Texas Louisiana continental shelf from the current-meter, nearsurface drifting buoy (Chu et. al 00b), propagation of long Rossby waves at mid-depths (around 00 m) in the tropical North Atlantic from the Argo float data (Chu et al. 00), and temporal and spatial variability of the global upper ocean heat content (Chu 0) from the data of the Global Temperature and Salinity Profile Program (GTSPP, Sun et al. 00). The spectral mode truncation is the key for the success of the method. It acts as a spatial low pass filter for the fields to allow the highest wavenumbers corresponding to the highest spectral eigenvalues without aliasing due to the information provided from the observational network. Questions arise: Can a simple and effective mode truncation method be developed to take into account of model resolution (i.e., total number of model grid points)? What are the major differences between and? What is the quality and uncertainty of the method? The purpose of this paper is to answer these questions. The remainder of the paper is organized as follows. Section describes error analysis. Section presents the steep-descending mode truncation method. Section shows idealized truth and observational fields. Section compares analysis fields between and. Section introduces three synoptic monthly gridded world ocean temperature, salinity, and absolute geostrophic velocity datasets produced with the method and quality controlled by the NOAA National Centers for Environmental Information (NCEI). Conclusions are given in Section. Appendices A and B briefly describe several methods to determine the H matrix. Appendix C shows the determination of basis

7 functions. Appendix D presents the Vapnik-Chervonkis dimension for mode truncation. Appendix E depicts a special B matrix for this study.. Error Analysis Low mode truncation does not represent the reality well, while high mode truncation may contain too much noise. Let the truncated spectral representation sk in () at the grid points form an N-dimensional vector, T K K 1 K K N The M-dimensional innovation vector [see ()] s s ( r), s ( r ),..., s ( r ). () T (1) () ( M) d d( r ), d( r ),..., d( r ) at observational points can be transformed into the grid points M ( m) hnmd M m1 n ( n), n nm fn m1 D D r f h, (1) where D(rn) represents the observational innovation at the grid points, D( r ) c ( r ) c ( r ). (1) n o n b n From Eq(a), observations at grid points are computed using co(rn) = H T co(rm). The original background state, cb(rn), keeps in the grid space. The matrix form of (1) is T FD H d, (1) where fn denotes contribution of all observational data unto the grid point rn. The larger the value of fn, the larger the observational influence on that grid point (rn). D is an N-dimensional vector at the grid points, T D (,,..., ) (1) D1 D D N

8 The analysis error (i.e., analysis ca versus truth ct) in the spectral data assimilation [see ()] is given by a( rn) ca( rn) ct( rn) s ( r ) D( r ) ( r ) c ( r ) c ( r ) c ( r ) c ( r ) c ( r ) c ( r ) a n b n o n b n o n t n K n n o n Here, () and (1) are used. The analysis error is decomposed into two parts (0) ( r ) ( r ) ( r ), (1) a n K n o n with the truncation error given by ( r ) s ( r ) D( r ), (a) K n K n n and the observational error given by ( r ) c ( r ) c ( r ). (b) o n o n t n. Steep-Descending Mode Truncation The Vapnik-Chervonkis dimension (Vapnik 1; Chu et al. 00a, 0) was used to determine the optimal mode truncation Kopt. As depicted in Appendix D, it depends only on the ratio of the total number of observational points (M) versus spectral truncation (K) and does not depend on the total number of model grid points (N). This method neglects observational error and ignores the model resolution. In fact, the analysis error variance over the whole domain is given by where E M ε Fε = ε Fε ε Fε ε Fε, ε Fε e, () N T T T T T a a a K K K o o o o o o e o is the observational error variance [see ()]. Here, the observational error is assumaed the same at grid points as at the grid points. This is due to the simplification of the error covariance matrix R = eo I. The Cauchy-Schwarz inequality shows that

9 T T T T Ea εkfεk εkfεk εofεo εofεo E E M / Ne ( M / N) e K K o o The relative analysis error reduction at the mode-k can be expressed by the ratio E E M / Ne Me / N ln, K,,... E E M Ne Me N K1 K1 o o K K K / o o / Both EK and EK-1 are large for small K (low-mode truncation), which may lead to a small value of γk. Both EK and EK-1 are small for large K (high-mode truncation), which also leads to a small value of γk. An optimal truncation should be between the low-mode and high-mode truncations with a larger value (over a threshold) of γk. This procedure is illustrated as follows. The values (γ, γ,, γkb) are calculated using () from a large KB (say 0). The mean and standard deviaition of γ can be computed as, K B K () () 1 K B K 1 B K, s= ( K ). () 1 K B Suppose that the relative error reductions (γ, γ,, γkb) satisfy the Gaussian distribution. An K 0(1 - α)% upper one-sided confidence bound on γ is given by zs, () th which is used as the threshold for the mode truncation. Here, z is the random variable satisfying the Gaussian distribution with zero mean and standard deviation of 1. If several γ values exceed the threshold, the highest mode KOPT max( K) () Kth is selected for mode truncation. After the mode truncation KOPT is determined, the spectral coefficnets (ak, k = 1,,, KOPT) can be calculated, and so as the truncation error variance E K OPT.

10 Multi-Platform Observations Let observation be conducted by L instruments with different e ( l ) deployed at ( m r ) l (ml = 1,,.., ML; l = 1,,, L). The total number of observations is M observational vector is represented by c T o o L Ml. The M-dimensional (1) () ( M1) (1) () ( M) co( r1 ), co( r1 ),..., co( r1 ), co( r ), co( r ),..., co( r ),..., (1) () ( M L ) () co( rl ), co( rl ),..., co( rl ) The observational error variance is given by εfε M [ e ] M [ e ]... M [ e ]. () T (1) () (L) o o 1 o o L o The relative error reduction γk for mode truncation () is replaced by E E M / Ne M ( e ) / N ln, K,,... E E M Ne M e N L L ( l) ( l) K1 K1 l o l o l1 l1 K L L ( l) ( l) K K l / o l( o ) / l1 l1 After the mode truncation is determined, the equation (1) is used to get the analysis field.. Truth, Background, and Observational Fields Consider an artificial non-dimensional horizontal domain (-1 < x < 1, - < y < ) with the four curved rigid boundaries (Fig. ): x y x / (west) 0.cos sin / (east) () y x y / (south) 0.sin 1 cos / (north) The domain is discretized with Δx = Δy = 0.. The total number of the grid points inside the domain (N) is. Fig. shows the first 1 basis functions {ϕk}, which are the eigenvectors of the Laplacian operator with the Dirichlet boundary condition, i.e., b1 = 0 in (C) of Appendix C. l1 l (1)

11 The first basis-function 1 ( x n) shows a one-gyre structure. The second and third basis functions ( ) xn and ( ) xn show the east-west and north-south dual-eddies. The fourth basis- function ( ) x n shows the east-west slanted dipole-pattern with opposite signs in the northeastern region (positive) and the southwestern region (negative). The fourth basis-function ( ) x n shows the tripole-pattern with negative values in the western and eastern regions and positive values in between. The higher order basis functions have more complicated variability structures. Two truth fields for the non-dimensional domain with rigid and curved boundaries (Fig. ) contain multiple mesoscale eddies (treated as truth ) given by ct( x, y) y /cos Lx( x, y) sin Ly( x, y) x y x y x y 0.cos sin, 0.sin 1 cos, () Lx, Ly,,, / for the large-eddy field (Fig a) and given by ct( x, y) y /cos Lx( x, y) cos Ly( x, y) x y x y x y 0.cos sin, 0.sin 1 cos Lx, Ly,,,0 for the small-eddy field (Fig. b). The background field is given by b (, ) / () c x y y () The observational points {r (m) } are randomly selected inside the domain (Fig. ) with the total number (M) of 0. The observational points {r (m) } are kept the same for all the sensitivity studies. The domain is discretized by Δx = Δy = 0. with total number (N) of grid points of,.

12 Sixteen sets of observations (co) are constructed from Figs.a, b using the analytical values plus white Gaussian noises (εo) of zero mean and various standard deviations (σ) from 0 (no noise) to.0 with 0.1 increment from 0 to 1.0 and 0. increment from 1.0 to.0 (total 1 sets), generated by the MATLAB, c r c r r. () ( m) ( m) ( m) o( ) t( ) o( ) Figs. a and b show out of the 1 constructed sets with σ = (0, 0., 0.,., 1.,.0). Both and methods are used to get the analysis field ca(rn) from these observations. The bilinear interpolation (see Appendix B) is used for the observation operator H in this study.. Comparison between and a. Analysis Fields The steep-descending mode truncation KOPT depends on the user-input parameter eo [see ()] and observational noise σ. E a and γk are computed from the observational data in Figs. a and b. The threshold of mode truncation () varies with the significance level α. In this study, (e0, σ) vary between 0 and ; α has two levels of (0.0, 0.) with z0.0 = 1., z0. = 1. in (). For given values of e0 (= 0.) and σ (= 0.), the optimal mode truncation depends on the significance level α with KOPT = for α = 0.0 (Fig. a) and KOPT = for α = 0. (Fig. b). Most results shown in this section is for α = 0.0 since it it a commonly used significance level. For the large-eddy field, KOPT is not sensitive to the values of σ and eo. It is in the upper-left portion and in the lower-right portion of Table 1. For the small-eddy field, KOPT takes (, ) for the most cases, 1 for the high noise levels ( 1. ) and low eo values ( eo 1.0 ), and for the low noise levels ( 0.1) and low eo values ( eo 0. ) (Table ). 1

13 The analysis field using the data assimilation (1) for a particular user-input parameter eo and noise level σ, c ( r, e, ), is represented in Fig. a (the large-eddy field) using a n o observations in Fig. a (with various σ), and in Fig. b (the small-eddy field) using observations in Fig. b (with various σ). Comparison between Figs. a, b and Figs. a,b demonstrates the capability of the method with the analysis fields c ( r,, e ) fully a n o reconstructed for all occasions. b. Analysis Fields With the assumption that the c field is statistically stationary and homogeneous, the equation () with the R and B matrices represented by () and (E1) [see Appendix E] is used to analyze the observational data with three user-defined paramters: (ra, rb, eo). Here, ra and rb are the decorrelation scale and zero crossing (rb > ra); eo is the standard deviation of the observational error. Let these paramters take discrete values with total number of Pa for ra, Pb for rb, and Pe for eo. In this study, we set Pa = Pb = Pe =. eo has values (0., 0., 1.0, 1.,.0). Considering the horizontal domain from - to in both (x, y) directions, ra takes values (,,,, ); (rb - ra) takes values (0., 1.0, 1.,.0,.). There are combinations of (ra, rb, eo) for the test. The analysis field from the data assimilation (), c ( r,, r, r, e ), with four different a n a b o sets of user-input parameters (ra, rb - ra, eo): (,., 1), (,., 1), (,., 1), and (,., ), are presented in Fig. a (the large-eddy field) using observations in Fig. a, and in Fig. b (the small-eddy field) using observations in Fig. b. Comparison between Figs. a, b and Figs. a,b demonstrates strong dependence of the output on the selection of the parameters (ra, rb, eo). For the large-scale eddies (Fig. a), the analysis fields ca are very different from the truth field ct for ra =, rb =., eo = 1 for all observations (Fig. a); the difference between the 1

14 reconstructed and truth fields decreases as ra and rb increase; the two fields are quite similar when ra =, rb =. for both eo = 1 and. Such similarity reduces with increasing eo. For the small-scale eddies (Fig. b), the analysis fields ca are totally different from the truth field ct for ra =, rb =., eo = 1 and for all observations (Fig. b), less different as ra and rb decrease; and are quite similar to ct when ra =, rb =., eo = 1. c. Root Mean Square Error The analysis field from, c a, depends only on the observational error variance e o and its uncertainty is represented by the root mean square error R, 1 R (, e ) c (,, e ) c( ) N o a n o t n N r r. (a) n1 Average over all the values of eo leads to the overall uncertainty The analysis field using ( c a 1 R ( ) c (,, e ) c ( ) NP uncertainty due to a particular parameter is represented by N a rn o t r n. (b) e eo n1 ) depends on three user-defined parameters (ra, rb, eo). Its 1 R (, r ) (,, r, r, e ) ( ) N a a n a b o t n NPbP r r, (a) e rb eo n1 1 R (, r ) (,, r, r, e ) ( ) N b a n a b o t n NPaP r r, (b) e ra eo n1 1 R (, e ) (,, r, r, e ) ( ) N o a n a b o t n NPaP r r, (c) b ra rb n1 which are compared to R ( ) and R (, e o ).

15 Fig. shows the comparison between R (, r a ) and R ( ) for different ra values: (,,,, ) and two types (the large-scale and small-scale) of the observational field. R (, r a ) monotonically increases with σ and is generally larger than R ( ). For the observations representing the large-scale eddy fields (Lx =, Ly =, see Fig. a), R ( ) increases slightly from 0. for σ = 0 to 0. for σ =.0. However, R (, r a ) is always larger than R ( ) and increases from 0. for σ = 0 to 1.1 for σ =.0; R (, r a ) is smaller than R ( ) for small σ, equals R ( ) at certain σ0, and larger than R ( ) for σ > σ0. The value of σ0 increases with ra from 0. for ra = to 1.0 for ra =. R (, r a ) increases from 0.1 for σ = 0 to 0. for σ =.0. For the observations representing the small-scale eddy field (Lx =, Ly =, see Fig. b), R ( ) increases slightly from 0. for σ = 0 to 0. for σ = 0.; evidently from 0. for σ = 0. to 0. for σ = 0.; and slowly from 0. for σ = 0. to 0.1 for σ =.0. However, R (, r a ) is much larger than R ( ) for any ra. For example, R (, r a ) increases from 0. for σ = 0 to 1. for σ =.0;, R (, r a ) increases from 0. for σ = 0 to 1.0 for σ =.0. Fig. shows the comparison between R (, r b ) and R ( ) for different (rb ra) values: (0., 1.0, 1.,.0,.) and two types (large-scale and small-scale) of the observational fields. R (, r b ) monotonically increases with σ and is generally larger than R ( ). For the observations representing the large-scale eddy fields (Lx =, Ly =, see Fig. a), R (, r r ) monotonically increases with σ from around 0. for σ = 0 to around 0. for σ = b a.0 for all the values of (rb ra) with σ0 from 0. for (rb ra) = 0. to 0. for (rb ra) =.. For the observations representing the small-scale eddy fields (Lx =, Ly =, see Fig. b),

16 R (, r r ) is much larger than R ( ) for any (rb -ra) and σ. For example, b a R (, r r 0.) increases from 0. for σ = 0 to 1.00 for σ =.0;, R (, r r.) b a increases from 0. for σ = 0 to 1.00 for σ =.0. Fig. 1 shows the comparison between R (, e o ) and R (, e o ) for different eo values: (0., 0., 1.0, 1.,.0) and two types (large-scale and small-scale) of the observational fields. First, R (, e o ) monotonically increases with σ and is evidently larger than R (, e o ) for all σ and eo. Second, dependence of R (, e o ) on σ is insensitive to the change of eo. For the observations representing the large-scale eddy fields (Lx =, Ly =, see Fig. a), R (, e o ) is close to R (, e o ) for σ < 1., and much larger than R (, e o ) for σ > 1. with eo = 0. and 0.; and vice versa with eo = 1.0, 1., and.0. R (, e o.0) increases slightly from 0. at σ = 0 to 1.0 at σ =.0 and is almost twice of R (, e o ) for all σ. For the observations representing the small-scale eddy fields (Lx =, Ly =, see Fig. b), R (, e o ) is also larger than R (, e o ). For example, R (, e o.0) increases slightly from 1. at σ = 0 to 1. at σ =.0, which is - times of R (, e o.0) for σ < 1.0. The overall performance between and with various noise levels (σ) can be estimated by the error ratio, R ( ) ˆ 1 ( ), R ( ) c (,, r, r, e ) c ( ) ˆ R ( ) NP P P a b o N a rn a b o t r n. () a b e r r e n1 Fig. 1 shows the dependence of κ(σ) (evidently less than 1) on σ for the two types (large-scale and small-scale eddies) of the observational fields represented by Figs a and b with two different significance levels (α = 0.0, 0.) for the threshold of mode truncation in the 1 b a

17 method (). At α = 0.0 (Fig. 1a), for the large-scale eddy field, κ(σ) takes 0.1 at σ = 0; fluctuates with σ; and decreases to 0. at σ =.0. For the small-scale eddy field, κ(σ) increases monotonically with σ from 0. at σ = 0 to 0. at σ =.0. At α = 0. (Fig. 1b), for the largescale eddy field, κ(σ) takes 1.1 at σ = 0; decreases monotonically with σ to 0. at σ =.0. For the small-scale eddy field, κ(σ) increases monotonically with σ from 0. at σ = 0 to 0.0 at σ =.0. It means that the performs better for the test case. Integration of κ(σ) over the whole interval of the noise level [0,.0] yields ˆ ( ) d large-scale eddy () small-scale eddy which means that the overall error for the is % (1%) of the error for the large-scale (small-scale) eddy field for α = 0.0. The overall performance of the method is relatively insensitive to the selection of the significance level α. The computational cost of the and methods is comparable in the test cases. In the method, the steepdescending method for mode truncation requires (a) the computation of a large number Kb in Eq() of eigenvectors, (b) the construction and solution of the equation (1) can be done once for all. In the method, however, the construction and solution of the equation () must be repeated each time background/observations changes.. Synoptic Monthly Gridded Temperature and Salinity Fields The method is used to to produce the synoptic monthly gridded (SMG) temperature (T) and salinity (S) datasets (Chu and Fan 01a, Chu et al. 01) from the two world ocean observational (T, S) profile datasets [the NOAA national Centers for Environmental Information (NCEI) s World Ocean Database (WOD) and the Global Temperature and Salinity Profile Program (GTSPP)]. The synoptic monthly gridded absolute geostrophic velocity dataset (Chu 1

18 and Fan 01b) is also established from the SMG-WOD (T, S) fields using the P vector method (Chu 1, Chu and Wang 00). These datasets have been quality controlled by the NCEI professionals and are openly downloaded for public use at &f=searchpage. The duration is January 1 to December 0 for the synoptic monthly gridded WOD (T, S) and absolute geostrophic velocity fields and January 10 to December 00 for the synoptic monthly gridded GTSPP (T, S) fields.. Conclusions Ocean spectral data assimilation has been developed on the base of the classic theory of the generalized Fourier series expansion such that any ocean field can be represented by a linear combination of the products of basis functions (or called modes) and corresponding spectral coefficients. The basis functions are the eigenvectors of the Laplace operator, determined only by the topography with the same lateral boundary condition for the assimilated variable anomaly. They are pre-calculated and independent on any observational data and background fields. The mode truncation K depends on the observational data and a user input parameter e o (i.e., observational error variance); and is determined via the steep-descending method. The completely changes the common ocean data assimilation procedures such as, KF, and variational methods, where the background error covariance matrix B needs to be pre-determined since the weight matrix W is used. However, the uses the spectral form to represent the observational innovation at the grid points [see (1)]. Minimization of the truncation error variance leads to the optimal selection of the spectral coefficients. Thus, the background error covariance matrix B vanishes in the procedure since the weight matrix 1

19 W is not used. It is contrast to the existing method, where the B matrix is often assumed to be stationary and homogeneous with user-defined parameters. The capability of the method is demonstrated through its comparison to using analytical D fields of large and small mesoscale eddies inside a domain with rigid and curved boundaries as truth, and addition to the truth of white Gaussian noises with zero mean and standard deviations (σ) varying from 0 (no noise) to.0 with 0.1 increment at randomly selected locations used as observations. A simple covariance function (Bretherton, 1) was used for the procedure with three user-defined parameters (ra, rb, eo) taking possible values each. The uses the same value of eo. The performance of and is compared by (1) patterns for each set of combinations of parameters, () root mean square errors for varying parameters, and () overall root mean square erros. The results show that the overall error reduction using the is evident, which is % (1%) [% (%)] for significance level α = 0.0 (α = 0.) of the error for the large-scale (small-scale) eddy field. In context of practical application, synoptic monthly gridded world ocean temperature, salinity, and absolute geostrophic velocity datasets have been produced with the method and quality controlled by the NOAA National Centers for Environmental Information (NCEI). Two issues need to be addressed on the correlation matrix. First, the comparison between the and is at one particular instant in time. The B matrix used in the is based only on distance. Second, in the covariancematrix based methods, when the covariance matrix is fixed once and for all, it is wellknown that the very first data assimilation cycle is doing well, but subsequent cycles are less effective because the remaining error has a tendency to be orthogonal to the directions of the covariance matrix. In the method, the correction is based on spectral functions (i.e. basis functions) chosen onceandfor all. More sophisticated, flowbased 1

20 covariance matrix will allow to perform much better. Further verification and validation under real-time ocean conditions are needed to verify the quality of in time cycles and to compare between and methods. In the two test cases (large and small eddy fields), it is clear that the optimal mode truncation KOPT (around for the large eddy field and around 0 for the small eddy field) are very closed to the number of eigenvectors required to represent the truth field (Fig. ). This shows the capability of the steep-descending mode truncation. However, the performance of the method for the truth field is a mixture of large and small scales in different parts of the domains needs to be further investigated. Acknowledgments. The Office of Naval Research, the Naval Oceanographic Office, and the Naval Postgraduate School supported this study. 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 hmn in H-matrix are influenced more by the nearby points and less by the more distant points. Let d ( x x ) ( y y ) (A1) m ( m) ( m) n i j be the distance between the grid point (xi, yj) and observational point (x (m), y (m) ). The influence of the grid point xn on the observational point x (m) is given by (Spepard 1) N m q m / mn n n n1 where q is an arbitrary positive real number called the power parameter (typically, q = ). Another form of hmn is given by (Franke and Nielson ) 0 q h d d (A)

21 h mn [( D d ) / D d ] N n1 ( m) m ( m) m n n [( D d ) / D d ] ( m) m ( m) m n n, (A) where D (m) is the distance from the observational point x (m) to the most distant grid point. Eq.(A) has been found to give better results than (A). As a result, cb(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. B-1), x, i x xi 1 yj y yj 1. Mathematically, the variable cb at r (inside the grid cell) can be represented approximately by a polynomial, L L b i j 0 0 c () r A ( x x)( y y ) (B1) where L = 1 refers to the bilinear interpolation, and L = leads to the bicubic interpolation. For the bilinear interpolation, Eq.(B1) becomes c ( r ) A A ( xx ) A ( y y ) A ( xx )( y y ) (B) b 00 i 01 j i j or in matrix notation, A00 A 1 01 c () r b 1 ( xxi) A ( y y ) A. (B) j Since cb at four neighboring grid points: cb(xi, yj), cb(xi+1, yj), cb(xi+1, yj), cb(xi+1, yj+1) are given, substitution of the four values into (B) leads to the determination of the four coefficients A00, A, A01, A. Using these coefficients, the bilinear interpolation (B) becomes 1

22 c ( x, y ) c ( x, y ) c x x y y x x y y b i1 j1 b i1 j b( r) ( i)( j) ( i)( j1 ) ( xi 1xi)( yj 1 yj) ( xi 1xi)( yj 1 yj) c ( x, y ) c ( x, y ) ( x x)( y y ) ( x x)( y y) ( x x )( y y ) ( x x )( y y ) b i j1 b i j i1 j i1 j1 i1 i j1 j i1 i j1 j Let the observational point r (m) be located in the grid cell, ( m) x, i x xi 1 ( m) yj y yj 1. Evaluation of cb at the observational point r (m) using (B) leads to c ( r ) p c ( x, y ) p c ( x, y ) p c ( x, y ) p c ( x, y ) (B) ( m) ( m) ( m) ( m) ( m) b i, j b i j i1, j b i1 j i, j1 b i j1 i1, j1 b i1 j1 ( m) ( m) ( m) ( m) where the proportional coefficients { p,, p 1,, p, 1, p i j i j i j i 1, j 1 } are defined by p ( m) i, j1 p ( m) i, j ( m) ( m) i1 j1 ( x x )( y y ) ( x x )( y y ) i1 i j1 j ( m) ( m) ( xi 1 x )( y yj) ( x x )( y y ) i1 i j1 j ( m) or in matrix notation, p, i1, j1 It is noted that the proportionality coefficients { location of the observational points (r (m) ), and p ( m) i1, j i, j i1, j i, j1 i1, j1 ( m) ( m) i j1 ( x x )( y y ) ( x x )( y y ) ( m) ( m) i i1 i j1 j ( x x )( y yj) ( x x )( y y ) i1 i j1 j ( m) ( m) ( m) ( m) p, p, p, p i, j i1, j i, j1 i1, j1,. (B) (B) } depend solely on the ( m) ( m) ( m) ( m) p p p p 1. (B) Setting L = in (B1) leads to the bicubic spline interpolation, c ( r) A A ( x x ) A ( y y ) A ( x x )( y y ) b 00 i 01 j i j 0 ( i) 0 ( j) ( i) A x x A y y A x x 1( i) ( j) 1 ( i)( j) 0( j) A x x y y A x x y y A y y (B)

23 b ) 1 ( i) ( i) ( i) A A A A A A A ( ) 1 0 y y j A ( ) 0 A1 0 0 y yj A ( ) y y j ( r (B) c x x x x x x which is rewritten by 1 xi xi x i 0 1 xi xi cb ( r) 1 x x x xi A00 A A0 A0 1 A A A 0 y y 1 j A A1 0 0 yj y j y A yj yj y j y 1 (B) Determination of the ten coefficients (A00, A01, A0, A0, A, A, A1, A0, A1, A) requires not only the values,, A00 cb( xi, yj), 00 0 b i1 j A A x A ( x) A ( x) c ( x, y ) A00 A01y A0( y) A0( y) cb( xi, yj 1), ( ) 0( ) A A x A y A x y A x A y ( ) 1( ) 1 ( ) 0( ) A x A x y A x y A y c ( x, y ), b i1 j1 but also the derivatives at the neighboring grid points A c ( x, y ) / x [ c ( x, y ) c ( x, y )] / x, b i j b i1 j b i1 j, A c ( x y )/ y [ c ( x, y ) c ( x, y )]/ y, 01 b i j b i j1 b i j1 A ( x) A ( x) A c ( x, y ) / x 0 b i1 j [ c ( x, y ) c ( x, y )]/ x, b i j b i j (B)

24 b i j1 A ( y) A ( y) A c ( x, y )/ x [ c ( x, y ) c ( x, y )]/ x, b i1 j1 b i1 j1 A ( x) A ( x) A c ( x, y )/ y 01 1 b i1 j [ c ( x, y ) c ( x, y )] / y, b i1 j1 b i1 j b i j1 A ( y) A ( y) A c ( x, y ) / y [ c ( x, y ) c ( x, y )] / y. b i j b i j (B1) The solution of the above set of linear algebraic equations (B) and (B1) leads to the determination of the ten coefficients (A00, A01, A0, A0, A, A, A1, A0, A1, A). It is noted that values of cb at the neighboring grid points (xi, yj), (xi+1, yj), (xi, yj+1), (xi+1, yj+1), (xi-1, yj), (xi, yj-1), (xi+, yj), (xi-1, yj+1), (xi+1, yj-1), (xi, yj+) are used to solve (B) and (B1). Following (B), interpolation of cb at the neighboring grid points on the observational r (m) [=(x (m), y (m) ] using the bi-cubic interpolation is given by 1 xi xi x i ( m) ( m) ( m) ( m) 0 1 xi xi cb ( r ) 1 x ( x ) ( x ) xi A A A A ( m) A A A1 0 y j y ( m) A A1 0 0 yj y j ( y ) ( m) A yj yj y j ( y ) (B1) Thus, an equation similar to (B) can be written for evaluating cb at the observational point r (m) with the known coefficients (A00, A01, A0, A0, A, A, A1, A0, A1, A, ( m) ( m) ( m) ( m) b r i, j b i j i1, j b i1 j i, j1 b i j1 c ( ) p c ( x, y ) p c ( x, y ) p c ( x, y ) ( m) ( m) ( m) ( m) i1, j1 b i1 j1 i1, j b i1 j i, j1 b i j1 i, j b i j p c ( x, y ) p c ( x, y ) p c ( x, y ) p c ( x, y ) ( m) ( m) i1, j1 b i1 j1 i1, j1 b i1 ( m) j1 i, j b i j p c ( x, y ) p c ( x, y ) p c ( x, y ) (B)

25 ( m) ( m) ( m) ( m) where the corresponding coefficients { p,, p 1,, p, 1, p i j i j i j i 1, j 1 p ( m) i, j (r (m) ), and, ( m) i 1, j p, ( m) i, j 1 p, ( m) i, j p, ( m) i 1, j 1 p, ( m) i 1, j 1 p, } are analytically determined and depends solely on the location of the observational points ( m) ( m) ( m) ( m) ( m) ( m) ( m) ( m) ( m) ( m) p p p p p p p p p p 1 (B) i, j i1, j i, j1 i1, j1 i1, j i, j1 i, j i1, j1 i1, j1 i, j Since only neighboring grid points are used to interpolate at the observational point r (m) using the bicubic interpolation, the matrix H has only 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. (0), three necessary conditions should be satisfied in selection of basis-functions { k () r }: (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 cb) and the basis functions { k }. The second necessary condition is given by where δkk is the Kronecker delta, k() r k' () r dr kk', (C1) kk ' 0 if k k'. (C) 1 if k k' 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 { k } of the Laplacian operator with the same boundary condition as the variable anomaly (c cb),

26 k k k, b1 ( ) e k b ( ) k 0, k 1,...,. (C) Here, {λk} are the eigenvalues, e is the unit vector normal to the boundary; τ denotes a moving point along the boundary, and [ b 1 (), b () ] are parameters varying with τ. The boundary condition in (C) becomes the Dirichlet boundary condition when b1 = 0, and the Neumann boundary conditions when b = 0. As pointed by Chu et al. (0), different variable anomalies have different [ b 1 (), b () ]. For example, the temperature, salinity, and velocity potential anomalies have b = 0 for the rigid boundary and b1 = 0 for the open boundary. However, the anomaly has b1 = 0 for the rigid boundary and b = 0 for the open boundary. It is obvious that the eigenvectors { k } are orthonormal and independent of the assimilated variables. Appendix D. Vapnik-Chervonenkis Dimension for Mode Truncation The Vapnik-Chervonenkis dimension (Vapnik 1; Chu et al. 00a, 0) is to seek the optimal mode truncation on the base of the first term of the analysis error (), with the cost function Here, α ( J J ( K, M, ), K tr J N ( K ) [ fn( Dn Dn )] T tr εkfε K n1 (D1) N 1 ln( M / K) 1 ln( / M) ( KM,, ) J*, K1,,..., M / K (D) 1) is the significance level. J* is the upper bound of Jtr. For a given M, Jtr decreases monotonically with K; μ increases with K if α is given. The optimal mode truncation is through the minimization of the cost function, min ( J ) J. (D) K K K opt

27 This method neglects observational error [only first term of () 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 Kopt. 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. (1) proposed B r ij r ij b ij, bij 1 exp, r, N N ij i j rb r a r b r r r, (E1) a depending on distances only. Here, rij is the distance between the two grid points ri and raj; ray and rb are the decorrelation scale and zero crossing. To conduct the data assimilation, the three parameters (eo, ra, rb) need to be defined by user. Chu et al. (1, 00) compute autocorrelation functions from historical observational data to fit the Gaussian function and get decorrelation scales for the B matrix. Recent studies show that some variables such as upper ocean current speed does not satisfy the normal distribution, but the Weibull distribution (Chu 00, 00). References Bretherton FP, Davis RE, Fandry CB (1) A technique for objective analysis and design of oceanographic experiments applied to MODE-. Deep-Sea Res. Oceanogr. Abstr.,,, doi:.1/00-1() Chu (1) P-vector method for determining absolute velocity from hydrographic data Chu PC (00) Probability distribution function of the upper equatorial Pacific current speeds. Geophys Res Lett,, doi:./00gl0.

28 Chu PC (00) Statistical characteristics of the global surface current speeds obtained from satellite altimeter and scatterometer data. IEEE J Sel Topics Earth Obs Remote Sensing, (1) -. Chu PC (0) Global upper ocean heat content and climate variability. Ocean Dyn, 1 (), -. Chu PC, Wells SK, Haeger SD, Szczechowski C, Carron M (1) Temporal and spatial scales of the Yellow Sea thermal variability. J Geophys Res (Oceans),, -. Chu PC, Wang GH, Chen YC (00) Japan/East Sea (JES) circulation and thermohaline structure, Part, Autocorrelation Functions. J Phys Oceanogr,, -. Chu PC, Wang GH (00) Seasonal variability of thermohaline front in the central South China Sea. J Oceanogr,, -. Chu PC, Ivanov LM, Korzhova TP, Margolina TM, Melnichenko OM (00a, b) Analysis of sparse and noisy ocean current data using flow decomposition. Part 1: theory. J Atmos Oceanic Technol, 0, 1; Part : application to Eulerian and Lagrangian data. J Atmos Oceanic Technol, 0, 1. Chu PC, Wang GH Fan CW (00a) Evaluation of the U.S. Navy's Modular Ocean Data Assimilation System (MODAS) using the South China Sea Monsoon Experiment (SCSMEX) data. J Oceanogr, 0, 0-1. Chu PC, Ivanov LM, Margolina TM (00b) Rotation method for reconstructing process and field from imperfect data. Int J Bifur Chaos,, 1. Chu PC, Ivanov LM, Melnichenko OM (00a) Fall-winter current reversals on the Texas- Louisiana continental shelf. J Phys. Oceanogr.,, 0. Chu PC, Ivanov LM, Margolina TM (00b) Seasonal variability of the Black Sea Chlorophyll-a concentration. J Mar Syst,, 1. Chu PC, Ivanov LM, Melnichenko OV, Wells NC (00) Long baroclinic Rossby waves in the tropical North Atlantic observed from profiling floats. J Geophys Res,, C00, doi:./00jc00. Chu PC, Tokmakian RT, Fan CW, Sun LC (0) Optimal spectral decomposition () for ocean data assimilation. J Atmos Oceanic Technol,, -. Chu PC, Fan CW (01a) Synoptic monthly gridded three dimensional (D) World Ocean Database temperature and salinity from January 1 to December 0 (NCEI Accession 01). NOAA National Centers for Environmental Information (NCEI),

29 Chu PC, Fan CW (01b) Synoptic Monthly Gridded WOD Absolute Geostrophic Velocity (SMG-WOD-V) (January 1 - December 0) with the P-Vector Method (NCEI Accession 0). NOAA National Centers for Environmental Information (NCEI), Chu PC, Fan CW, Sun LC (01) Synoptic monthly gridded Global Temperature and Salinity Profile Programme (GTSPP) water temperature and salinity from January 10 to December 00 (NCEI Accession 01). NOAA National Centers for Environmental Information (NCEI), Cohn SE (1) Estimation theory for data assimilation problems: Basic concenptual framework and some open questions. J Meteor Soc Japan,, -. Evensen G (00) The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn,, -. Franke R, Nielson G () Scattered data interpolation and application: a tutorial and survey. In Hagen H., Roller D. (eds) Geometric modelling, methods and applications. Berlin, Springer, -10. Galanis GN, Louka P, Katsafados Kallos PG, Pytharoulis I (00) Applications of Kalman filters based on non-linear functions to numerical weather predictions. Ann Geophys,, 1-0. Han GJ, Wu XR, Zhang SQ, Liu ZY, Li W (01) Error covariance estimation for coupled data assimilation using a Lorenz atmosphere and a simple pynocline ocean model. J Climate,, 1-1. Ide K, Courtier P, Ghil M (1) Unified notation for data assimilation: Operational, sequential and variational. J Meteor Soc Japan,, -1. Oke PR, Brassington GB, Griffin DA, Schiller A (00) The Bluelink ocean data assimilation system (BODAS). Ocean Modelling, 1, -0. Spepard D (1) A two-dimensional interpolation function for irregularly spaced data. Proc rd Nat Conf ACM, 1-. Sun LC, Thresher A, Keeley R, et al. (00) The data management system for the Global Temperature and Salinity Profile Program (GTSPP). In Proceedings of the OceanObs 0: Sustained Ocean Observations and Information for Society Conference (Vol. ), Venice, Italy, 1 September 00, Hall, J, Harrison D.E. and Stammer, D., Eds., ESA Publication WPP-. Tang Y, Kleeman R (00) SST assimilation experiments in a tropical Pacific ocean model. J. Phys. Oceanogr.,, -.

30 Vapnik, VH (1) Reconstruction of Empirical Laws from Observations (in Russian). Nauka, pp. Yan CX. Zhu J, Xie JP (0) An ocean data assimilation system in the Indian Ocean and west Pacific Ocean. Adv Atmos Sci, doi:.0/s z.

31 Table 1. Dependence of KOPT on (σ, eo) for the large-eddy field shown in Fig. a with significance level α = 0.0. eo σ Table. Dependence of KOPT on (σ, eo) for the small-eddy field shown in Fig. b with significance level α = 0.0. σ eo

32 Figure Captions Fig. 1. Illustration of ocean data assimilation with cb located at the grid points, and co located at the points *. The ocean data assimilation is to convert the innovation, d = co Hcb, from the observational points to the grid points. Fig.. Horizontal non-dimensional domain with four curved rigid boundaries with each boundary given by Eq. (). Fig.. Basis functions from ϕ1 to ϕ1 for the domain depicted by Eq.(). Fig.. Truth field ct taken as (a) the analytical function () with large-scale eddy field Lx=, Ly =, β = π/, and (b) the analytical function () with small-scale eddy field Lx =, Ly =, β = 0. Fig.. Randomly selected locations (total: 0) inside the domain as observational points. Fig. a. Observational data (co) from Fig. a with added white Gaussian noises of zero mean and various standard deviations: (a) 0 (i.e., no noise) (b) 0., (c) 0., (d) 1.0, (e) 1., and (f).0. Fig. b. Observational data (co) from Fig. b with added white Gaussian noises of zero mean and various standard deviations: (a) 0 (i.e., no noise) (b) 0., (c) 0., (d) 1.0, (e) 1., and (f).0. Fig.. Dependence of E a and γk on K for the observational data for the small-scale eddy field with σ = 0. and eo =0. at two significant levels of (a) α = 0.0 (z 0.0 = 1.) and (b) α = 0. (z 0. = 1.1) as the threshold of mode truncation [see Eq.()]. The optimal mode truncation is for α = 0.0 and for α = 0.. Fig. a. The analysis field ca obtained by the spectral data assimilation [see Eq.(1)] using the steep-descending mode truncation with the significance level of α = 0.0 from the observations shown in Fig. a with noise (σ) levels (0, 0., 0., 1.0, 1.,.0) and values of eo: (a) 0., (i.e., no noise), (b) 0., (c) 1.0, and (d).0. Fig. b. The analysis field ca obtained by the spectral data assimilation [see Eq.(1)] using the steep-descending mode truncation with the significance level of α = 0.0 from the observations shown in Fig. b with noise (σ) levels (0, 0., 0., 1.0, 1.,.0) and values of eo: (a) 0., (i.e., no noise), (b) 0., (c) 1.0, and (d).0. Fig. a. The analysis field ca obtained by the data assimilation [see Eq.()] for observations shown in Fig. a various noise levels with various combinations of user-defined parameters (ra, rb, eo,): (,., 1), (,., 1), (,., 1), and (,., ).

33 Fig. b. The analysis field ca obtained by the data assimilation [see Eq.()] for observations shown in Fig. b various noise levels with various combinations of user-defined parameters (ra, rb, eo,): (,., 1), (,., 1), (,., 1), and (,., ). Fig.. Comparison between R (, r a ) and R ( ) of the analysis fields from the same observations with different noise levels with varying parameter ra = (,,,, ) from top to bottom with the left panels using observations shown in Fig. a and the right panels using observations in Fig. b. The solid curves represent the with the significance level of α = 0.0; and the dotted curves refer to the. Fig.. Comparison between R (, r b ) and R ( ) of the analysis fields from the same observations with different noise levels with different (rb ra) = (0., 1.0, 1.,.0,.) with the left panels using observations shown in Fig. a and the right panels using observations in Fig. b. The solid curves represent the with the significance level of α = 0.0; and the dotted curves refer to the. Fig. 1. Comparison between R (, e o ) and R (, e o ) of the analysis fields from the same observations with different noise levels with varying parameter eo = (0., 0., 1.0, 1.,.0) from top to bottom with the left panels using observations shown in Fig. a and the right panels using observations in Fig. b. The solid curves represent the with the significance level of α = 0.0; and the dotted curves refer to the. Fig. 1. Dependence of the error ratio κ [see Eq.()] on σ using observations in Fig. a (represented by dots) and in Fig. b (represented by *) with two different significance levels: (a) α = 0.0, and (b) α = 0.. Fig. B1. Interpolation at an observational point r (m) from four neighboring grid points.

34 Fig. 1. Illustration of ocean data assimilation with cb located at the grid points, and co located at the points *. The ocean data assimilation is to convert the innovation, d = co Hcb, from the observational points to the grid points.

35 Fig.. Horizontal non-dimensional domain with four curved rigid boundaries with each boundary given by Eq. ().

36 Fig.. Basis functions from ϕ1 to ϕ1 for the domain depicted by Eq.().

37 Fig.. Truth field ct taken as (a) the analytical function () with large-scale eddy field Lx=, Ly =, β = π/, and (b) the analytical function () with small-scale eddy field Lx =, Ly =, β = 0.

38 Fig.. Randomly selected locations (total: 0) inside the domain as observational points.

39 Fig. a. Observational data (co) from Fig. a with added white Gaussian noises of zero mean and various standard deviations: (a) 0 (i.e., no noise) (b) 0., (c) 0., (d) 1.0, (e) 1., and (f).0.

40 Fig. b. Observational data (co) from Fig. b with added white Gaussian noises of zero mean and various standard deviations: (a) 0 (i.e., no noise) (b) 0., (c) 0., (d) 1.0, (e) 1., and (f).0.

41 Fig.. Dependence of E a and γk on K for the observational data for the small-scale eddy field with σ = 0. and eo =0. at two significant levels of (a) α = 0.0 (z 0.0 = 1.) and (b) α = 0. (z 0. = 1.1) as the threshold of mode truncation [see Eq.()]. The optimal mode truncation is for α = 0.0 and for α = 0..

42 Fig. a. The analysis field ca obtained by the spectral data assimilation [see Eq.(1)] using the steep-descending mode truncation with the significance level of α = 0.0 from the observations shown in Fig. a with noise (σ) levels (0, 0., 0., 1.0, 1.,.0) and values of eo: (a) 0., (i.e., no noise), (b) 0., (c) 1.0, and (d).0.

43 Fig. b. The analysis field ca obtained by the spectral data assimilation [see Eq.(1)] using the steep-descending mode truncation with the significance level of α = 0.0 from the observations shown in Fig. b with noise (σ) levels (0, 0., 0., 1.0, 1.,.0) and values of eo: (a) 0., (i.e., no noise), (b) 0., (c) 1.0, and (d).0.

44 Fig. a. The analysis field ca obtained by the data assimilation [see Eq.()] for observations shown in Fig. a various noise levels with various combinations of user-defined parameters (ra, rb, eo,): (,., 1), (,., 1), (,., 1), and (,., ).

45 Fig. b. The analysis field ca obtained by the data assimilation [see Eq.()] for observations shown in Fig. b various noise levels with various combinations of userdefined parameters (ra, rb, eo,): (,., 1), (,., 1), (,., 1), and (,., ).