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Correction to “A three‐dimensional variational data assimilation scheme for the Regional Ocean Modeling System: Implementation and basic experiments”

2008; American Geophysical Union; Volume: 113; Issue: C6 Linguagem: Inglês

10.1029/2008jc004928

2156-2202

Zhijin Li, Yi Chao, James C. McWilliams, Kayo Ide,

Climate variability and models

[1] In the paper “A three-dimensional variational data assimilation scheme for the Regional Ocean Modeling System: Implementation and basic experiments” by Z. Li et al. (Journal of Geophysical Research, 113, C05002, doi:10.1029/2006JC004042, 2008), several variables were printed incorrectly. The affected sections are reprinted below. [3] ROMS3DVAR executes data assimilation for each nested level. Because all levels use the same formulation, we focus in the subsequent discussions on L2, which has the smallest domain but the highest horizontal resolution. For consistency with the ROMS open boundary conditions, ROMS3DVAR takes the forecast and returns the analysis at all ROMS L2 model grid points but not at those determined by L1. [4] To handle the large-dimension background error covariance matrices while taking anisotropy and inhomogeneity into account, ROMS3DVAR computes a three-dimensional (3-D) correlation matrix as a Kronecker product Cξκ ⊗ Cη of a two-dimensional (2-D) matrix Cξκ in the vertical and cross-shore directions and a one-dimensional (1-D) matrix Cη in the alongshore direction [Li et al., 2008]. Here ξ, η, and κ stand for the alongshore, cross-shore, and vertical directions, respectively. ROMS3DVAR also computes the 2-D surface correlation matrix Cξη for nonsteric SSH as a Kronecker product, that is, Cξ ⊗ Cη. [5] Any Cξκ, Cξ, or Cη is grid-based, i.e., correlations are defined between two grid points. A difficulty arises, however, in constructing Cξκ using the S-coordinate in the presence of variable topography. To illustrate this, let us consider an error correlation Cξκ(i1k1, i2k1) between a nearshore grid point (i1k1) and a grid point (i2k1) 15 km offshore and on the same S-level k1 near the ocean bottom in two different vertical cross-shore sections. In a section south of Monterey Bay, the depth at (i1k1) is about 10 m and at (i2k1) about 500 m (Figure 1). Because the depth of thermocline and mixed layer in this region is about 30 m in late summer and less than 100 m in winter, Cξκ(i1k1, i2k1) between these two 2-D grid points should be near zero. In a section north of the bay, however, Cξκ(i1k1, i2k1) may not necessarily be zero because the depth of the grid point (i2k1) is much shallower than 500 m. This variation in the correlations presents a fundamental difficulty in constructing Cξκ using the S-coordinate. [6] To circumvent the difficulty, the ROMS3DVAR grid employs a Z-coordinate for the vertical discretization. The same staggered ROMS C-grid is used in the horizontal so that the resulting analysis will easily satisfy the lateral boundary condition. In this study the ROMS3DVAR grid uses Nξ × Nη = 82 × 178 horizontal curvilinear grid points, and Nκ = 24 vertical Z-levels in L2. [7] Accordingly, the execution of the incremental 3DVAR given a ROMS forecast on the S-coordinate is accomplished through the following procedure. Using spline interpolation [Akima, 1970], the background state on the Z-coordinate is first formed by transforming the ROMS forecast. After performing the incremental 3DVAR, the lateral boundary condition along the coastline is imposed on the analysis increment in the Z-coordinate. ROMS3DVAR produces the final analysis by transforming and adding the analysis increment to the original ROMS forecast on the S-coordinate. The transformation back to the S-coordinate is performed on the analysis increment (not the full analysis) to reduce the interpolation errors and thus help maintain the delicate dynamical balance that is attained by forward model integration in the ROMS forecast. A concern may arise, however, concerning whether the final analysis satisfies the lateral boundary condition on the S-coordinate. To address this concern, we examined the final analysis in various experiments and found that the lateral boundary condition is, in general, adequately satisfied. [8] For the estimation of coastal ocean dynamics with timescales ranging from hours to days, we adopted for ROMS3DVAR a 6-h assimilation cycle. The first cycle of the day begins by performing a 6-h ROMS forecast using the analysis valid at 0300 UTC as an initial condition; 0300 UTC corresponds to 7 pm local standard time (LST). Once the valid 0900 UTC 6-h ROMS forecast is completed and all observations in the 6-h time bin between 0600 and 1200 UTC are collected, ROMS3DVAR executes the incremental 3DVAR and computes the analysis valid at 0900 UTC by treating all observations as if they were taken at 0900 UTC. This completes the first 6-h cycle. ROMS3DVAR repeats the cycle four times a day and produces analyses valid at 0300, 0900, 1500, and 2100 UTC. [9] Having described the general ROMS-DAS procedure and its grid arrangement, we turn our attention to the issues concerning estimation of the standard deviation matrices and the correlation matrices for the background errors. This section presents a general method for estimation of the standard deviation and correlation matrices associated with Bζ′, Bψ″χ″, and BTS, along with its application to ROMS3DVAR. [10] Having constructed Σξηκ, we now present a method for estimating the self-correlation Cξηκ from the surrogate data: Cξηκ represents any of Cζ′ζ′ξη, Cψ″ψ″ξηκ, Cχ″χ″ξηκ, CTTξηκ, or CSSξηκ, though the vertical correlation are suppressed for the 2-D correlation matrix Cζ′ζ′ξη. Because ROMS3DVAR uses a Kronecker product-based algorithm Cξηκ = Cξκ ⊗ Cη to construct the self-correlation matrix [Li et al., 2008], we describe a method for computing Cξκ and Cη. We also present a method for estimating the error cross-correlation matrix CTSξηκ between temperature and salinity. Although the current ROMS3DVAR neglects the error cross-correlation between ageostrophic streamfunction and velocity potential, it can be incorporated by considering the cross-correlation Cψ″χ″ξηκ. [11] To compute the ROMS3DVAR self-correlation matrices, we assume that the correlations from the surrogate data have the same structure as the forecast errors. This allows us to compute Cξκ and Cη for the five self-correlations directly from the surrogate data. [13] If the surrogate data set has a large enough sample, then the resulting correlations are locally smooth between neighboring grid points. However, spurious correlations can occur between remote grid points, leading to undesirably long tails in Cξκ and Cη [Gaspari and Cohn, 1999]. Such long tails can cause noisy analysis increments. A common technique to address this problem is to apply a localization function that retains correlations computed from the surrogate data within a local neighborhood but suppresses all correlations at large distances. For localization of Cξκ and Cη, we use a Gaussian matrix. The final correlation matrix is then given by the Hadamard (or Schur) product of a correlation matrix computed from the surrogate data and the Gaussian matrix. [14] When using the Hadamard product to construct a correlation matrix, caution should be exercised: The Hadamard product produces a positive definite matrix only if both matrices are positive definite [e.g., Horn and Johnson, 1994]. While the correlation matrix estimated from the surrogate data is generally positive definite, the Gaussian matrix is not necessarily so, especially for large length scales. In ROMS3DVAR a Cholesky factorization is applied to the correlation matrices for preconditioning [Li et al., 2008, section 4.1]. The use of the LAPACK code for the Cholesky factorization acts as a verification of the positive definiteness of Cξκ and Cη [Anderson et al., 1999]. [15] For the localization of Cξκ and Cη in ROMS3DVAR, we use a Gaussian matrix with a horizontal length scale of 50 km and a vertical length scale of 400 m. For the nonsteric SSH, the 2-D correlation Cζ′ζ′ξη(i1j1, i2j2) is described by Li et al. [2008]. The decorrelation length scale of Cζ′ζ′ξ is about 10 km nearshore, increases to about 35 km further offshore and remains the same beyond 80 km offshore. Here the decorrelation length scale is estimated as the distance to the point where the correlation reduces to e−1. In contrast, Cζ′ζ′η is basically homogeneous with the decorrelation length of about 30 km, although the decorrelation length scale decreases slightly around the latitude of Monterey Bay. [16] Figure 7 shows the 2-D correlations Cξκ(i1k1, i2k2) of ageostrophic streamfunction Cψ″ψ″ξκ and ageostrophic velocity potential Cχ″χ″ξκ, for fixed (i1k1) at either an offshore or a nearshore location and varying (i2k2) in the vertical cross-shore section. The property of the 1-D alongshore correlations Cψ″ψ″η and Cχ″χ″η are similar to those of Cζ′ζ′η, that is, they are basically homogeneous with the decorrelation length of about 30 km. [17] A slight difference between Cψ″ψ″ξκ and Cχ″χ″ξκ appears in Figure 7 in the decorrelation length scale: Cψ″ψ″ξκ has slightly larger vertical but smaller cross-shore decorrelation length scale than Cχ″χ″ξκ. Otherwise Cψ″ψ″ξκ and Cχ″χ″ξκ have quite similar spatial patterns. In general, the cross-shore decorrelation length scale decreases as the distance from the coast increases, while the vertical decorrelation length scale does not change in the cross-shore direction. [18] Figure 8 shows the 2-D correlation Cξκ(i1k1, i2k2) of temperature CTTξκ for (i1k1) fixed at offshore and nearshore locations at depths of 50 m and 200 m. The behavior of CTTξκ is quite similar to that of Cψ″ψ″ξκ and Cχ″χ″ξκ, but the decorrelation length scales of CTTξκ are slightly larger in the cross-shore direction and smaller in the vertical direction than those of Cψ″ψ″ξκ and Cχ″χ″ξκ. Moreover, the cross-shore variation of the vertical decorrelation length scale is fairly small. Variations in the decorrelation length scale, smaller near the surface, can be explained by the dominance of baroclinic normal modes. The properties of the 1-D alongshore correlation CTTη are similar to those of Cζ′ζ′η, as well as those of Cψ″ψ″η and Cχ″χ″η. [19] The vertical cross-shore correlation of salinity CSSξκ(i1k1, i2k2) is shown in Figure 9. While decorrelation length scales are very similar to those of CTTξκ(i1k1, i2k2), the slant structure of CSSξκ(i1k1, i2k2) nearshore suggests a strong influence of the thermocline variability as observed in ΣSS(ξκ (Figure 6). The larger vertical decorrelation length scales nearshore for salinity are understandable. They reflect the fact that, during upwelling, salinity near the surface is controlled by the amount of upwelled deep water. This feature contributes to the significance of the inseparability in the cross-shore and vertical directions. The properties of the 1-D alongshore correlation CSSη are similar to those of Cζ′ζ′η and CTTη. [20] From Figures 8 and 9, we can see the variation of the vertical decorrelation length scale with depth. The vertical decorrelation length scale becomes larger with depth. This variation is also similar for the ageostrophic streamfunction and velocity potential, though their vertical decorrelation length scales are larger overall than those of temperature and salinity. [21] In ROMS3DVAR, the cross-correlation CTSξηκ between temperature and salinity is parameterized by the local cross-correlation vector rTS and the average of the corresponding self-correlations CTTξηκ and CSSξηκ [Li et al., 2008, equation (24)]. Figure 10 shows rTS at the surface and at a depth of 75 m. [22] SSHs are observed with satellite altimetry and tide gauges. SSH observations have been a major data source for ocean data assimilation owing to, not only wide availability, but also their ability to constrain the temperature and salinity profiles through the hydrostatic balance [e.g., Cooper and Haines, 1996]. A weak hydrostatic balance constraint is essential for coastal data assimilation. As shown in Figure 3 for the standard deviation, the nonsteric SSH dominates steric SSH near the coast. This suggests that the use of a strong constraint is unreasonable there. In contrast, the steric SSH dominates in the offshore region. This implies that without the weak or strong constraint, the background error covariance matrix must include the cross-correlations between total SSH, temperature, and salinity. These cross-correlations are inhomogeneous and have long vertical decorrelation length scales. It is difficult to reconcile such complex relations in the cross-correlations in a way that is computationally feasible for an incremental 3DVAR algorithm. [23] In the absence of other types of observations, a SSH observation induces analysis increments in the three ROMS3DVAR control variables, δxζ′a, δxTa, and δxSa, by invoking the weak constraint for hydrostatic balance. Geostrophic balance is, however, applied as a strong constraint. Thus, the velocity increment is strictly determined by δxζ′a, δxTa, and δxSa. [24] Figure 11 shows the analysis increment using a single SSH observation at an offshore location with a positive innovation, i.e., the value of the observation is higher than that of the ROMS3DVAR forecast. Both steric and nonsteric SSH increments are positive, leading to geostrophic velocity increments that have an anticyclonic eddy-like structure. The positive steric SSH increment is related to a positive temperature increment and a negative salinity increment. An anisotropic effect of the decorrelation length scale in the error correlations is visible in the length scales of the increments that are larger in the alongshore direction. If the observation were taken exactly at a ROMS3DVAR grid point, then the nonsteric SSH increment would have a similar spatial pattern to the horizontal error correlation Cξη = Cξ ⊗ Cη that assumes horizontal separability [Li et al., 2008, Figure 3]. Therefore, an observation function can act as a smoother when the corresponding observation is not taken at a ROMS3DVAR grid point. [25] The vertical spread of the observed information is shown in Figure 12 for the same experiment. The increments are mainly contained above a depth of 150 m. The maxima of the temperature and salinity increments are centered at about 50 m, which is around the bottom of the mixed layer. Both geostrophic velocity increments have their maximum at the surface but penetrate deeper than the temperature and salinity increments. This is due to the barotropic pressure gradient induced by the nonsteric SSH increments. [26] To examine the sensitivity to observation location, we carry out another experiment with a single nearshore observation. The analysis increments are shown in Figure 13. All increments have smaller length scales than in the offshore experiments due to the cross-shore inhomogeneity of the decorrelation length scales (section 3.3.2). [27] The steric SSH increment is much less dominant in comparison with the offshore observation. The steric SSH increment accounts for more than 80% of the total SSH increment in the offshore case, while it accounts for only about 30% in the nearshore case. The non-steric SSH increment is more important than the steric SSH increment nearshore, which is consistent with the ratio of the nonsteric and steric error variances as can be inferred from Figure 3. [28] In the absence of other types of observations, a velocity observation induces analysis increments in the four ROMS3DVAR control variables, δxψ″a, δxχ″a, δxTa, and δxSa, through the weak constraint for the geostrophic balance. Total velocity increments have both geostrophic and ageostrophic components and hence depend on all four control variables. The relative amplitude of the geostrophic and ageostrophic increments depends on the relative amplitude of the corresponding standard deviations: in general, ageostrophy is significant near the coast while geostrophy rules in the offshore region. The SSH increment is completely determined by δxTa and δxSa because hydrostatic balance is applied as a strong constraint. [29] Figure 14 shows the analysis increment using a single observation of alongshore velocity at a nearshore location with a positive innovation. The increment of the total cross-shore velocity shows a classic “butterfly-like” structure. One interesting result is the relative value of the geostrophic velocity and ageostrophic velocity. The geostrophic velocity accounts for about 60% of the total. [30] Temperature is the most observed variable in the ocean. In the absence of other types of observations, a temperature observation induces analysis increments in two control variables, δxTa and δxSa; the latter is caused by the cross-correlation CTSξηκ between them. Both hydrostatic balance and geostrophic balance are applied as strong constraints; hence δxTa and δxSa completely determine the SSH and velocity increments. [31] Figure 15 shows the analysis increment for a single observation of temperature at a nearshore location with a positive innovation. The increment structures of the velocity components are classic ones. The increment of salinity is obviously due to the cross-correlation. Consistent with Figure 10, the cross-correlation is negative nearshore, and thus the increments of salinity and temperature are opposite in sign. [32] The AOSN experiment in Monterey Bay, California, August 2003, provided an unprecedented number of in situ observations by a variety of means and instruments; details of the experiment can be found at the Web site (http://www.mbari.org/aosn). During August 2003, ROMS3DVAR generated analyses and forecasts in real-time using the 6-h forecast cycle (section 2.2). After the experiment, additional observations became available while some of the original observations were upgraded. The results presented in this section are obtained using all observations available. The observations include the NAVOCEANO MCSST Level 2 High Resolution Picture Transmission and Local Area Coverage (HRPT/LAC) 2.2 km sea surface temperature data set (level 2), underwater glider temperature/salinity profiles, aircraft SSTs, ship temperature/salinity/depth (CTD), and Automatic Underwater Vehicle (AUV) temperature/salinity profiles (http://www.mbari.org/aosn for details about observations). [33] In this reanalysis experiment we adjusted the weak geostrophic balance constraint. The geostrophic velocity is computed after smoothing is applied to δxζ′, δxT, and δxS. The smoothing is a spatial average with the weight determined by the Gaussian function where d is the distance and r0 = 3 km (approximately two times the model grid size). This additional smoothing is used to reduce an overestimation of the geostrophic velocity at small scales due to the small decorrelation length scales nearshore. Also, it is a practical consideration. It has been shown that a minimization process generally acts first on the larger scales. Thus, the small scales are dealt with mainly during the last stage of the minimization process [Veerse and Thepaut, 1998]. In this experiment, we allow a maximum of 40 iterations for the minimization. In this case, it is possible that the minimization is terminated before it converges to the minimum of the cost function. In this case, some small-scale noise may remain. The smoothing can help reduce such small-scale noise. [34] Figure 10. Local cross-correlation rTS between the temperature and salinity (left) at the surface and (right) at a depth of 75 m. [35] Figure 15. Analysis increments for temperature, salinity, and geostrophic velocity components at the sea surface using a single temperature observation at a nearshore location at 122.2°W and 36.8°N. The innovation is 1.3°C. The standard deviation of the observation error is 0.35°C. The units are °C for temperature, psu for salinity, and meters per second for velocity.