## 1. Introduction

In mapping geophysical fields and their uncertainties, it is important to assess the sensitivity to parameters. The present work carries out such studies for the a priori error subspace estimation and mapping methodology introduced in (Lermusiaux et al. 2000, henceforth LAL00). With this three-dimensional, multivariate, and multiscale approach (appendix A), the a priori error subspace, that is, the dominant components of the a priori missing variability or probability density functions of the state to be mapped, are constructed in two parts. The observed portions of the subspace are specified based on measurements and multiscale statistical models. The nonobserved portions are then built by dynamical adjustments to the observed portions. Once this a priori error subspace is computed, the datasets of interest can be mapped into gridded fields and error properties by Bayesian estimation in the subspace. Presently, error statistics are assumed to be described by covariance matrices and the mapping (analysis) reduces to a minimum error variance estimation problem. The subspace is then determined by the dominant eigendecomposition of the a priori error covariance.

Two main objectives are to investigate (i) the sensitivity of the state and error estimates to the properties of the subspace (e.g., size, dynamics, adjustment) and (ii) the sensitivity of the subspace itself to its size and to the data and dynamics utilized in its construction. The types of questions considered include: How many, and why so many, eigenvectors should be employed to represent most of a covariance matrix or to carry out a mapping? How sensitive is the subspace to the parameters of the model employed in its dynamical adjustment? By how much should such subspaces be dynamically adjusted? What is the performance of forecasts generated from these analyses? Results obtained from different a priori parameters are compared among each other and also to those of the univariate horizontal (2D) scheme of the Harvard Ocean Prediction System (HOPS; Lozano et al. 1996; Robinson 1996). Since this benchmark is multiscale, the present 3D mapping of fields and error covariances is referred to in short as the “multivariate 3D” scheme. Ultimately, forecasts generated from the various analyses are compared to in situ and satellite data, using root-mean-square differences and correlation coefficients for measures of skill. The third objective is to extract dynamical knowledge from the variability subspaces and fields computed. This is carried out by studying the spatial distribution of the estimated hydrographic and velocity variances, shapes and amplitudes of dominant multivariate eigenvectors, and structures of (cross)-covariance functions.

The applications illustrated are in the Levantine Sea and focus on the mesoscale to subbasin-scale physical errors and fields. The measurements were collected during the LIW95 campaign (Roether et al. 1996; Malanotte-Rizzoli et al. 1996) carried out from January to April 1995 in the eastern Mediterranean. To introduce this region and its contemporary upper-thermocline features, a multivariate 3D estimate of the potential density at 105 m and velocity at 5 m is plotted in Fig. 1. Because of thermal-wind effects, several surface circulation structures are easily distinguished along the steepest slopes of isopycnals. To date, most investigations in the Levantine that combine data and dynamics have mainly resolved the subbasin scales and are usually based on close-to-geostrophy hypotheses (e.g., Hecht et al. 1988; Milliff and Robinson 1992; Özsoy et al. 1993; Robinson and Malanotte-Rizzoli 1993; Malanotte-Rizzoli et al. 1999).

Among the various techniques for the analysis of geophysical fields, only some estimate uncertainties (LAL00 and references cited therein). The present work is concerned with determining accurate a priori (error) covariance or structure functions and with estimating their parameters and covariance matrix form. Recent studies include: in atmospheric science, Rabier et al. (1998), Riishojgaard (1998), Dee and da Silva (1999), Gaspari and Cohn (1999), Gneiting (1999), Franke and Barker (2000); in physical oceanography, Cressie and Huang (1999), Miller and Cornuelle (1999), Menemenlis and Chechelnitsky (2000), Molinari and Festa (2000); and, in biological oceanography, Abbott and Letelier (1998). Today, error models, in particular the efficient representation of error covariance matrices, are essential for error predictions (Ehrendorfer 1997; Lermusiaux 1997, henceforth LER97) and data assimilation (Robinson et al. 1998; Chin et al. 1999; Derber and Bouttier 1999; Lermusiaux 1999a,b; Lermusiaux and Robinson 1999; Verron et al. 1999; Mitchell and Houtekamer 2000; Tippett et al. 2000; Hamill et al. 2001; Reichle et al. 2002).

In what follows, section 2 outlines a simple rule for estimating the size of error subspaces. In section 3, variations of these subspaces with their size and with the scales in the data, and the impacts of these variations on a posteriori fields, are illustrated for temperature and salinity. Section 4 proceeds to the complete physics: in addition to size and scales, nonlinear effects are studied and some dynamical properties derived. In section 5, the performance of forecasts initialized based on the present mapping is evaluated. Conclusions are in section 6. In the appendixes, algorithms are summarized and relevant nondimensional numbers listed.

## 2. The size of error subspaces: Back-of-the-envelope calculation

A simple rule giving an approximate order of magnitude for the size of multivariate error subspaces is outlined. It is based on experience with more quantitative calculations (e.g., Fukumori and Malanotte-Rizzoli 1995; LER97; Lermusiaux 1999a,b) and on the often red spectra of ocean phenomena. It is exemplified for the case where errors are mesoscale variability.

Consider first one state variable, for example, temperature or zonal velocity. For the vertical variability, two to three empirical orthogonal functions (EOFs) or dynamical modes often explain most of the significant variance (e.g., De Mey and Robinson 1987; Haney et al. 1995; De Mey 1997; Wunsch 1997; Pedder and Gomis 1998; von Storch and Frankignoul 1998). For the horizontal variability, most studies to date focus on surface fields. In that case, the number of dominant patterns or modes usually varies from 15 to about 50: for example, consider a surface field in the Gulf Stream or Atlantic region (e.g., Thacker and Lewandowicz 1996, 1997; Everson et al. 1997; Kaplan et al. 1997). For the 3D variability, an approximate subspace size is simply the product of vertical and horizontal requirements. With the above values, this product yields a size of 30–150.

^{1}of HOPS and the above mesoscale values, an approximate size is then 135–650. This is of the order of 10

^{2}to 10

^{3}. In summary, the order of magnitude of the size

*p*of an error subspace is often well approximated by,

*n*

^{υ}

_{h}

*n*

^{υ}

_{z}

*υ.*Similarly, a range of likely

*p*is obtained from the products of the minimum and maximum horizontal and vertical requirements; that is, Σ

_{υ}min(

*n*

^{υ}

_{h}

*n*

^{υ}

_{z}

*p*≤ Σ

_{υ}max(

*n*

^{υ}

_{h}

*n*

^{υ}

_{z}

The above sizes are much larger than classic 1D requirements but much smaller than the number of discrete state variables. For example, consider a nondimensional (from here on denoted by *) covariance matrix 𝗕* for a horizontal, univariate field. For covariance function, we use a “Mexican hat”^{2} and for discretized domain, the northwestern Levantine with a uniform resolution of 10 km (500 km in the *x* and 520 km in the *y* direction). The size of 𝗕* is thus (2703 × 2703). Computing its cumulative eigenvalue spectrum for the subbasin-scales (200-km zero crossing, 100-km decay scale), 15 eigenvectors represent 85% of the variance, 36 vectors 99%, and 50 vectors 99.8%. For the mesoscales (60-km zero crossing, 30-km decay scale), larger numbers are required: 15 eigenvectors represent 20% of the variance, 50 vectors 56%, and 100 vectors 83%. For at least 90% of the variance, 19 vectors are necessary for the subbasin-scales and 128 for the mesoscales. These sizes represent only 0.7% and 4.7% of the number of discrete state variables, 2703, respectively. Illustrations and further details are in LLA98.

## 3. Multivariate three-dimensional analysis of tracer fields: Subspace size

Hydrographic data (dynamical tracers *T* and *S*) are frequent observations. The sensitivity of the a posteriori state **x**^{a}_{trc}_{trc} is “observed” in the sense of LAL00, that is, historical (synoptic) data are available to specify its dominant eigendecomposition ^{p}_{trc}^{p}_{trc}

The data are 146 *T* and *S* profiles (Figs. 2a,b). Their error model (appendix A) includes a decorrelation time term. The numerical domain (Fig. 2c) contains 2703 horizontal grid points and 20 vertical levels (108 120 tracer state variables). There are 5840 data points [dimension of **y**^{o}, Eq. (A1b)], that is, the number of *T,* *S* scalar data residuals on vertical levels.

The benchmark univariate 2D scheme of HOPS assumes that two independent scales are present and that errors are homogeneous horizontally. It uses a successive correction, minimum error variance technique (e.g., Daley 1991; Lorenc 1992): the largest scales are gridded first and the resulting fields constitute the background in the mapping of smaller scales. Successively for each scale, global horizontal analyses of scale-filtered *T* and *S* data are thus carried out at various levels. For fair comparisons with this benchmark, the present multivariate 3D scheme is also a two-scale correction (*w* = 1, 2 in appendix A sections b–c), with horizontally homogeneous a priori errors. It thus carries out two multivariate 3D mappings successively. For each scale, ^{p}_{trc}**y**^{o}. For both schemes, the first a priori estimate **x**^{b}_{trc}*w* = 0 in appendix A sections b–c) is the horizontal average of **y**^{o}. Subsequently, the two scales are the subbasin-scale and mesoscale. At the mesoscale stage (section 3b), a priori errors are mesoscale variability but a posteriori fields are the total fields.

### a. Subbasin scale

Univariate 2D benchmark:Figure 3 shows four subbasin-scale (*w* = 1) univariate 2D analyses. The a priori horizontal covariance was set to be a “Mexican hat” (section 2) of 200-km zero crossing and 100-km decay scale, each estimated based on covariances among subbasin-scale data residuals. The data error covariance at data points was assumed diagonal (at each level, 𝗥 = *r* 𝗜) and of 50 days decorrelation timescale. The corresponding horizontally uniform, nondimensional (0 to 1) variance was calibrated to *r** = 0.03, mainly based on experience.

Multivariate 3D scheme:Figure 4 illustrates the construction of 𝗕_{trc} and Fig. 5 the mappings. The dominant eigendecomposition of ^{z}_{trc}*T,* *S* EOFs explains most of the variance (Fig. 4a). The horizontal covariance ^{r*}_{trc}*w* = 1) is finally carried out.

Figure 5a plots parts of the state **x**^{a}_{trc}*p* = 500 in Eq. (A16), that is, 500 eigenvectors are used, which explains 99.6% of the a priori error variance (Fig. 4b). The size of **x**^{a}_{trc}*T,* *S* maps are alike. For the subbasin scale, assuming zero vertical covariance and uncorrelated *T* and *S* error fields does not appear damaging. This is mainly because there is here more than enough data coverage for the subbasin scale. The vertical and *T*–*S* correlations are as expected stronger with the 3D scheme (Fig. 5a), but most differences are within data error bounds.

Comparing the *T,* *S* maps using 500 and 250 vectors (not shown), the relative root-mean-square (rms) differences are less than 1%. This is an interesting result. Once most (e.g., here ≥98%, see Fig. 4b) of the covariance has converged, the remaining eigenvectors can be ignored if their eigenvalues are negligible in comparison to data error eigenvalues (LLA98). Considering costs, the mapping with *p* = 250 is 10^{3} times cheaper than the full covariance mapping. Finally, using 10 vectors as on Fig. 5b is not sufficient. Indeed, the quick product of horizontal and vertical requirements for 99% of the variance each indicates here a total of 36 × 8 = 288 vectors [see Eq. (1) and use numbers in section 2 and Fig. 4a]. This back-of-the-envelope order of 10^{2} multivariate 3D vectors agrees with the numerically computed spectra of Fig. 4b.

### b. Mesoscale

Univariate 2D benchmark:Figure 6 shows four mesoscale (*w* = 2) univariate 2D analyses. Based on plots of covariances among data residuals, the zero crossing was set at 60 km, decay scale at 30 km, and decorrelation timescale at 7 days. Other parameters are as in section 3a.

Multivariate 3D scheme: In the mesoscale 3D mapping, **x**^{b}_{trc}_{trc} is illustrated by Fig. 7. In the vertical (Fig. 7a), the variance of the mesoscale data residuals [Eq. (A10)] decays less rapidly with the eigenvalue number than in the subbasin-scale case (Fig. 4a). In the horizontal, ^{r*}_{trc}*w* = 2) is finally carried out.

Figures 8a–b show **x**^{a}_{trc}*T,* *S* around 225 m, also seem better represented. For mesoscale mappings with data resolutions similar to those of Fig. 2a, a disadvantage of the univariate 2D analysis is in fact its sensitivity to environmental noise. Considering subspace sizes, the relative rms differences between maps of Figs. 8a and 8b are less than 3%. Even though 750 vectors explain 5% more a priori mesoscale variance than 500 vectors do (Fig. 7b), most of the additional 250 vectors have eigenvalues smaller than the local data error variance. They thus mainly yield mesoscale corrections within data error bounds.

For the 3D scheme, two main properties have been exemplified. First, once most of the a priori error covariance has converged (e.g., 99%), the remaining eigenvectors can be truncated in most cases. Second, the a priori error vectors of eigenvalues negligible with respect to data errors have negligible impacts on the a posteriori state. Even if the a priori error covariance has not converged, these vectors can usually be omitted.

## 4. Multivariate three-dimensional analyses of PE fields

Primitive equation fields and errors are now considered. Examples remain in the northwestern Levantine, but the period is now 19 March–16 April 1995. Four cases are discussed (Table 1): one univariate 2D benchmark and three variations of the multivariate 3D scheme.

The univariate 2D benchmark (case 1) first carries out horizontal analyses for *T* and *S,* as in section 3. Total velocity is then computed by assuming geostrophic balance with the gridded tracers and integrating the thermal wind equations up and down from a level of reference.

The present multivariate 3D scheme (cases 2–4) again uses a two-scale approach (*w* = 1, 2 in appendix A section b) to construct the a priori error subspace. For each scale, the tracer portion 𝗕_{trc} of the a priori error covariance 𝗕 is specified as in section 3. The nonobserved velocity portions are then built in dynamical accord with 𝗕_{trc}, through an ensemble of adjustment momentum integrations (tracers are kept fixed in these adjustments [see appendix A section b(2)]. These integrations lead to an ensemble of PE adjusted fields. Variability samples, that is, the differences between these fields and a priori state **x**^{b}, are then computed, normalized, and organized by singular value decomposition (SVD). Once a convergence criterion determines that the number of samples is large enough to explain most of the variance, an estimate 𝗕 ^{p} of 𝗕 is obtained. A mapping (appendix A section c) is then carried out.

The 3D cases of Table 1 were selected to illustrate the main tendencies in an extensive set of computations. They exemplify the sensitivities of the error subspace and state **x**^{a} to: (i) the nonlinear terms of the momentum equations used to adjust the nonobserved velocity variability to the observed tracer variability (cases 2 and 3), and (ii) the size of the subspace *p* (cases 3 and 4). Subbasin-scale estimates are only overviewed (section 4a). Mesoscale estimates are studied in more detail: a priori error covariances 𝗕 ^{p} = 𝗘**Π**𝗘^{T} (section 4b), a posteriori fields (section 4.3), and a posteriori error covariances 𝗕^{ap}^{a}**Π**^{a}𝗘^{aT}

### a. Subbasin scale

In each case (Table 1), the data **y**^{o} employed for the subbasin-scale are 512 profiles of the Mediterranean Ocean database (MODB) winter climatology for the Levantine (Brankart 1997, personal communication) and the *T*–*S* profiles for 10–18 February and 15 March (Fig. 2a). For coherence with the recent evolution of deep eastern Mediterranean waters (Roether et al. 1996), the MODB data were cut at 1250-m depth: only in situ data are utilized below 1250 m. In all cases, **x**^{b} is in hydrostatic equilibrium: its tracer fields are the horizontal averages of the data **y**^{o} and its velocity is null.

For the univariate 2D scheme (case 1), *T* and *S* fields are estimated from **y**^{o} as in section 3a. Total velocities are assumed in geostrophic balance, with a level of no motion at 600 m. For the multivariate 3D cases 2–4, the ^{p}_{trc}*p.* Three estimates of the subbasin-scale 𝗕 ^{p} are then built using the PE model, as indicated in Table 1. With these 𝗕 ^{p}'s, three mappings of the data **y**^{o} are carried out, following (appendix A section c).

### b. Mesoscale: Data and a priori error covariance

For each case (Table 1), the data for the mesoscale correction are 236 CTD profiles gathered from 19 March to 16 April (Fig. 9). They lead to 9440 data points (dimension of **y**^{o}). To have an acceptable synoptic coverage (see section 5), the field estimation is centered on 27 March. The data decorrelation timescale is adjusted to 7 days.

The four **x**^{b}'s (cases 1–4) are the four subbasin-scale fields of section 4a. Their a priori error is thus mesoscale variability: that is, for cases 2–4, **Γ** = 𝗶 in Eq. (A23). The corresponding covariances are computed using all 289 CTD profiles gathered from 10 February to 16 April (denoted by **y**^{o}_{ctd}

Univariate 2D benchmark: Based on plots of covariances among mesoscale residuals of the 289 CTD, the zero crossing, decay scale, and data decorrelation timescale are adjusted on basin average to 60 km, 30 km, and 7 days. Other parameters of the error covariance function for *T* and *S* (see Fig. 9b) are as in section 3a. Velocity error covariances are not computed.

Multivariate 3D scheme: In constructing the mesoscale 𝗕 ^{p} = 𝗘**Π**𝗘^{T}, the dominant decomposition of its tracer submatrix, ^{p}_{trc}_{trc}**Π**_{trc}^{T}_{trc}**y**^{o}_{ctdi}_{i}(**x**^{b}), *i* = 1, … , 289, Eq. (A7)] with the eigendecomposition of an analytical horizontal tracer covariance (of elements specified using the mesoscale Mexican hat of the benchmark). The resulting cumulative error variance is a curve similar to that of Fig. 7b. For the sizes *p* utilized (Table 1), the dominant 322 tracer eigenvectors explain 67.2% of the total tracer variance, the dominant 400, 72%, and dominant 500, 76.5%. The complete 𝗕 ^{p}'s are then obtained from ^{p}_{trc}*j* = 1, … , *p* dynamical adjustments lasted for 2 model days, with *p* increasing based on the convergence criterion *ρ* ≥ *α* [Eq. (A22)], with the limit *α* set to 0.93. This coefficient *ρ* ≤ 1 assesses the added value of new parallel batches of adjustment runs; some of its values are listed in Table 2a. For case 2, *ρ* ≥ *α* was achieved after 322 runs, for *p* = 322 and *p̃* = 288. For case 4, 400 runs were necessary, based on *p* = 400 and *p̃* = 360. This larger dimension is due to the nonlinear momentum terms that create flow variability not estimated by the linear adjustments of case 2. Since case 3 is simply the nonlinear extension of case 2, convergence is not an objective. Overall, the rate of increase of *ρ* diminishes as *p̃* ≤ *p* grows: this is partly because the singular values of 𝗕 decay rapidly.

Table 2b estimates the overall similarity among the three 𝗕 ^{p}'s, using Eq. (A22). Comparing cases 2 and 3, averaged on volume and state variables, nonlinear momentum terms only change about 2.7% of the nondimensional PE mesoscale variability. Since here **u**^{j} = **e**^{j} in Eq. (A17), the tracer covariance in a 𝗕 ^{p} is exactly ^{p}_{trc}*ρ*)*n*/*n*_{vel} = 5.4%, where *n* is the state vector size (218 943) and *n*_{vel} the number of velocity variables (110 823). This 5.4% difference measures the volume-averaged effect of nonlinear momentum terms on velocity covariances. Comparing cases 3 and 4, they are about 13.2% apart. The tracer eigenvectors of high number (small eigenvalues) here affect velocity covariances.^{3} Finally, comparing cases 2 and 4, their differences combine the nonlinear and size effects (2.7% and 13.2%).

The three 𝗕 ^{p} = 𝗘**Π**𝗘^{T}'s (cases 2–4) are now compared locally, focusing on variances [section 4b(1)], eigenvectors [section 4b(2)] and covariance functions [section 4b(4)]. These estimates substantially improve the univariate 2D scheme, which only provides 2D error variance fields.

#### 1) A priori mesoscale error variance

*Univariate 2D benchmark:* A priori tracer error variances are set to the horizontally averaged variance of the residuals of **y**^{o}_{ctd}

*Multivariate 3D scheme:* Error variances form the diagonal of the 𝗕 ^{p}'s. Tracer variances are horizontally uniform in the data domain (Fig. 9). For cases 2 and 3 (*p* = 322), the *T*–*S* error standard deviation at 5 m is 0.40°C/0.080 PSU, while at 500 m, it is 0.17°C/0.039 PSU. For case 4 (see LAL00), standard deviations are logically slightly larger: to first order^{4}, using the variance numbers given above, by 3.3%–3.6% [100 × (72 − 67.2)/2, divided by 72 and 67.2]. Total velocity variances are nonuniform in 3D. By construction, large a priori errors (Table 3) correspond to the locations and features estimated to have large mesoscale flow variability. They reflect local variations of the dominant density and momentum balances.

In case 2 (linear momentum), at the surface, the largest standard deviations are as expected above the Rhodes Basin (flanked by steep topography) within the deep, subbasin-scale Rhodes Gyre. Other features of above-average flow variability are, in decreasing order, the Ierapetra, Mersa Matruh Gyre (one of its lobes is in the domain) and central branch of the Mid-Mediterranean Jet. At 500-m depth, the largest *û,* *υ̂*

In case 3 (nonlinear momentum, *p* = 322), at the surface, it is now the high-velocity, small size Ierapetra that dominates. The correction to the larger Mersa Matruh Gyre is smaller, especially in *ψ.* For the subbasin-scale Rhodes Gyre and Mid-Mediterranean Jet, nonlinear terms only have a small impact. At 500-m depth, the two Straits and Rhodes Basin still dominate internal deviations, but case 3 also estimates that the Ierapetra and an anticyclone of the Mersa Matruh-Shikmona Gyre (MM-S-G) complex are above average. For these deep mesoscale vortices, nonlinear terms matter at depth. In case 4 (nonlinear momentum, *p* = 400), results are as those of case 3, except that standard deviations are a bit larger, to first order by about 6.6%–7.6% (13.2/2–13.2/2/0.868, see footnote 4).

Two general remarks can be made: (i) for accurate flow variability variances, not only the background density field, but also the regional topography, nonlinear, and diffusion terms should be considered; (ii) even though subspaces are used, the present global estimates of mesoscale variances are in accord with and complement previous local estimates (e.g., Özsoy et al. 1993; Robinson and Golnaraghi 1993).

#### 2) A priori mesoscale error eigenvectors and spectrum

*Case 2:*Figure 10 illustrates the first and second nondimensional singular vectors, columns of 𝗘* in Eqs. (A21)–(A23). They explain 1.08% and 1.05% of the variance explained by all 322 vectors, respectively. They are examined because several of their properties are representative of those of other dominant vectors. As expected from section 4b(1), for linear momentum, the variability in early spring is dominated by large-mesoscale variations of the Rhodes Gyre. In the last decades, the Rhodes Gyre has indeed been observed to be the main subbasin-scale feature of the region (Milliff and Robinson 1992; Nittis and Lascaratos 1998).

The first vector (Figs. 10, 1a–b) covers the principal core of the Gyre, above the Rhodes Basin. Horizontally (Fig. 10, 1a), the main *T* and *S* structures are almost axisymmetric, with horizontal decay scales of about 40 km (radius of 60 km). On the open-ocean side, the main *T,* *S,* and *ψ* structures indicate a predominant geostrophic equilibrium. Close to the coastline, this is not true mainly due to decreasing depth and increasing coastal friction. Vertically, *T* is surface intensified (Fig. 10, 1b), with a uniform extremum within 20–200 m and a significant extension down to 900–1000 m above the Rhodes Basin. The *S* structure has a subsurface extremum within 250 to 320 m and a zero crossing near 100 m. The signs of *T* and *S* reverse at about 1500 and 1300 m, respectively. Below, local extrema are much smaller than the global upper-thermocline extrema. Overall, where *T* and *S* are both relatively large, they are in phase, compensating each other in density. The surface *û* (Fig. 10, 1a) and cross sections in *û* and *υ̂**û* map and subsurface tracer cross sections. Its average amplitudes are about 20% of those of the main lobes. Vertically, *û* and *υ̂**ψ* (Fig. 10, 1a), a minimum total velocity is expected near 900 m.

The second vector (Fig. 10, 2a–b) mainly relates to displacements of the core of the Gyre, along the north–south axis of the Rhodes Basin. Except for the main double lobes in *T,* *S* and *ψ,* and corresponding triple lobes in *û* and *υ̂*

Other dominant vectors also relate to main features (Fig. 1 and Table 3). This is because to explain a natural variation, a group of vectors is usually required. Considering, for example, vectors 3–16, their largest amplitudes assign the vectors: (3–7, 9, 12–16) to the Rhodes Gyre; (3–6, 13–14) to wave patterns along the northern escarpment and Mid-Mediterranean Ridge; (5–9, 11–16) to the Ierapetra; (12–14) to the Mersa Matruh Gyre; (8–16) to the Mid-Mediterranean Jet; (3–4, 6, 9–16) to the West Cyprus Gyre; (4, 6, 9, 12–16) to the Asia Minor Current; and (9–10, 12–16) to the Mersa Matruh-Shikmona Gyre complex. Since each vector is orthogonal to all more dominant ones and 3D dynamical couplings can be complex, a detailed analysis requires additional computations beyond the present scope. Considering the decay of the error spectrum, the variance is distributed such that the dominant 50, 100, and 200 vectors explain 41.4%, 66.7%, and 89.4% of the total variance, respectively.

*Case 3:*Figure 11 is as Fig. 10, but for nonlinear momentum adjustments (Table 1). In accord with variance results (Table 3), the feature of dominant mesoscale variability is then the Ierapetra. The two vectors shown explain 1.59% and 1.24% of the variance explained by all 322 vectors, respectively. These percentages are larger than in case 2 because nonlinear terms steepen the start and flatten the upper middle of the eigenvalue spectrum. The first vector corresponds to the vortex itself, the second to eastward–westward displacements.

In comparison with case 2, horizontal *T* and *S* structures in the first vector (Fig. 11, 1a) are, when significant,^{5} closer to Mexican hats and of lesser width (radius near 50 km, decay scale of 30 km). Vertically (Fig. 11, 1b), *T* and *S* are more stretched than in case 2, as one expects for a ringlike vortex. Even though *T* remains surface intensified, its uniform extremum is deeper, within 50 to 250 m. This agrees with previous results: for example, the structure of Levantine warm core eddies (Brenner 1993) and shallow surface mixed layer of the Rhodes Gyre (e.g., Malanotte-Rizzoli et al. 1996). The dominant feature in *T* extends down to about 600 m, main depth of the present Ierapetra, instead of about 1000 m for the Rhodes Gyre (Fig. 10). The *S* structure has a zero crossing near 80 m and a subsurface extremum within 250–340 m. The signs in *T*/*S* reverse at about 1300 m/1100 m, which, as for the small extrema below, is shallower than at the Rhodes Gyre. Overall, *T* and *S* are still in phase when both important. The *ψ,* *û,* and *υ̂**û* and *υ̂**x* and *y* axes. The horizontal lobes (Fig. 11, 1a) present sharp gradients and are tilted, with their heads wider than their tails. Velocity is maximum near a radius of 30 km. Vertically (Fig. 11, 1b), *û* and *υ̂**ψ* (Fig. 11, 1a). The *û* and *υ̂*

Based on the second vector (Fig. 11, 2a,b), displacements of the Ierapetra can excite radiative momentum patterns. Such adjustments correlate most to density variations in thermal wind balance: note the *T* and *S* radiation patterns and horizontal *û* close to a “45 degree shamrock.” However, at all depths, *û* and *υ̂*

Since the two dominant vectors are modified, the others also differ, even though most of subspace itself remains unchanged (Table 2b). Considering again vectors 3–16, their largest amplitudes assign the vectors: (3–6, 8, 14–16) to the Ierapetra; (6–16) to the Rhodes Gyre; (4, 5, 8) to wave patterns along the northern escarpment and Mid-Mediterranean Ridge; (3–4, 7–10, 12) to the Mersa Matruh Gyre; (8–12, 14–16) to the Mid-Mediterranean Jet; (10–14, 16) to the West Cyprus Gyre; (10, 12, 16) to the Asia Minor Current; and (8–16) to the Mersa Matruh-Shikmona Gyre complex. Overall, the dominant 50, 100, and 200 vectors explain 41.1%, 66.2%, and 89.4% of the total variance, respectively. These percentages are close to those of case 2: the result is that here nonlinear terms impact the dominant vectors of the subspace but do not change much of the overall spectrum.

*Case 4:* The error subspaces of cases 3 and 4 were estimated to be similar at 86.8% (Table 2b). Up to an arbitrary sign, their vectors 1–15 are almost identical (vectors of case 4 are only slightly less noisy). Some differences start to occur by vector 16, mainly in the small mesoscales. The additional vectors 323–400 include 3D patterns of relatively large horizontal scales and usually “wiggly” vertical structures, or vice versa. The decay of the error spectrum is similar to that of case 3: the 50, 100, and 200 dominant vectors explain 38.4%, 62.7%, and 85.2% of the variance explained by all 400 vectors, respectively.

#### 3) Dynamical properties

From the dominant mesoscale vectors and fields obtained, the types and scales of possible wave patterns along the northern escarpment and Mid-Mediterranean Ridge (e.g., case 2), and the dynamical balance of the Ierapetra (e.g., cases 3–4) can be investigated. Note that the numbers obtained from cases 3 and 4 are similar, indicating a robustness of the results.

*Wave patterns:* Using app. B and Fig. 10, locally for a wave packet along the escarpment, *ρ*_{0} = 1028 kg m^{–3}, *L* = 25 km, *D* is about 1000 m, *H* is on Fig. 2c, *f*_{0} = 8.45 10^{–5} s^{–1}, and *β*_{0} = 1.86 10^{–11} m^{–1} s^{–1} (for *θ*_{0} = 35.5°). Computing a reference *ρ*_{s}(*z*) by horizontal average of the density field, this leads an average *N*_{s} within *D* of about 2.5 10^{–3} s^{–1}, a local *R*_{D} = *N*_{s}*D*/*f*_{0} ≃ 30 km, and *R*_{ext} ≃ 1200 km. For *U,* based on the second vector (Fig. 10, panels 2a, b), the sum of the surface internal and external components yields: *U* ≃ (3.9 × 1.5 10^{–2} + 5.6 10^{7} × 0.5 10^{–2}/*H*/*L*) (1893)^{1/2} ≃ 3 cm s^{–1} (1.5 10^{–2} and 0.5 10^{–2} are the nondimensional amplitudes, 3.9 and 5.6 10^{7} the normalization factors, and 1893 the eigenvalue). With contributions from other relevant vectors, the characteristic velocity *U* for a wave packet is about 5 cm s^{–1}. Hence, the local nondimensional numbers are *Ro* = 0.015, *Bu*_{ext} = 2 10^{3}, *Bu* = 1.4, *β* = 0.23, *α* ≃ 25, and *s* ≃ 0.6 (except just east of the island of Rhodes where *α* and *s* reach 50 and 1, respectively). The small *Ro* and large *Bu*_{ext} indicate that the rigid-lid linear PE regime is appropriate. From *β,* the planetary vorticity gradient is relatively small and from *α* ≫ 1, the topographic vorticity gradient is large: *β*_{0} can be neglected compared to |∇_{h}*H*|*f*_{0}/*D.* From *Bu* ∼ 1, the length scales of the horizontal flow and internal stretching (stratification's strength) match. Since *s* is close to one, quasigeostrophy may not apply. Finally, for the model topography (Fig. 2c), *s* and *α* are relatively uniform along the escarpment waveguide (about 60 km wide). Based on these numbers and interpretations, the topographic wave packets are of the coastal-trapped type (Huthnance 1992; Pedlosky 1987; Brink 1991). Note that here, wave patterns arise during rigid-lid momentum adjustments (fixed tracer disturbances): for the barotropic mode along the escarpment, to first-order in *s,* the frequency is then of the order (|∇_{h}*H*|*f*_{0}/*D*)/*k,* where *k* = 1/*L* is the wavenumber. This leads a period of about 1.45 days, that is, of the order of 1 to 2 days.

In the ocean, wave packets would be generated by complete dynamical adjustments, and also by winds, atmospheric pressure, and coastal flows. A complex wave spectrum around the above values can be expected. First, the natural topography is more jagged and at several locations steeper than the model topography, which we estimate could shift the period by about 50%. Second, several wave types are possible along the escarpment and/or Mid-Mediterranean Ridge: in addition to a suite of coastal-trapped waves (e.g., Leblond and Mysak 1971) like Kelvin and continental shelf waves, inertial and Rossby waves can be generated. For complete statements on scales, shapes, and forcings, additional studies are necessary (e.g., Huthnance 1978; Holland and Webster 1994). The dominance of such wave patterns and associated mesoscale variations in the vectors related to the Rhodes Gyre [section 4b(2)] suggests that they could be in part responsible for the observed multiple centers of the Gyre (Milliff and Robinson 1992). In particular, the main core of the Gyre is not stationary, but undergoes translations and precessions of synoptic timescales (seen in our simulations).

*Ierapetra:* Using Fig. 11 and local scale analysis, an approximate dynamical balance for the Ierapetra can be obtained. One has *ρ*_{0} = 1028 kg m^{–3}, *L* = 20 km, *D* = 400 m, *H* ≃ 2500 m (Fig. 2c), *f*_{0} = 8.15 10^{–5} s^{–1}, and *β*_{0} = 1.89 10^{–11} m^{–1} s^{–1} (for *θ*_{0} = 34.1°). The vertical averages within *D* of *N*_{s} and of the horizontal Δ*ρ* are 2.5 10^{–3} s^{–1} and 0.15 kg m^{–3}, respectively. Hence, *R*_{D} = 12 km and *R*_{ext} = 770 km. From Table 3, the mesoscale component of the total surface *U* is 25 cm s^{–1}. Adding the subbasin-scale component (section 4a) yields *U* ≃ 45 cm s^{–1}. Hence, for the Ierapetra, relevant nondimensional numbers are *Ro* = 0.28, *Bu*_{ext} = 1500, *Bu* = 0.38, *β* = 0.017, *α* = 1.8, and *s* = 0.5. The size of *Ro* attests that nonlinearities are important while *β* and *Bu*_{ext} confirm that the planetary vorticity and stretching of the free surface do not matter. The above numbers suggest that the dominant horizontal momentum balance within the vertical extent *D* is the gradient–wind balance: solving for *U* in *U*^{2}/*L* − *f*_{0}*U* + Δ*ρgD*/*ρ*_{0}*L* ≃ 0 gives a root *U* ≃ 50 cm s^{–1} (accounting for the uncertainty in characteristic scales, this root lies between 40 and 70 cm s^{–1}). In the vertical, since *Ro* and *Bu* are of the same order, *W* is large enough for the vertical convergence and horizontal divergence to balance (*W* ≃ *UD*/*L*) and the horizontal changes in density to match the vertical ones (Δ*ρ* ≃ Δ*ρ*_{s}). Based on *α* and *s,* the local topographic valley (Fig. 2c) is a factor in maintaining the location of the vortex. Finally, viscous terms are only important in the bottom layers and at the submesoscales: for examples, the horizontal momentum Ekman number *Ek*_{h} = *A*_{h}/*f*_{0}*L*^{2} is 0.92 for scales under 20 km but decays to 5 10^{–4} at 50 km and is almost null beyond 100 km (*A*_{h} is the effective Laplacian horizontal eddy viscosity, which is here scale dependent).

#### 4) A priori mesoscale error covariance functions

The covariance function between the surface (5 m) temperature at (33.80°N, 27.85°E: about 80 km east-southeast of the Ierapetra) and the other state variables is chosen as an example. The plots (Figs. 12–13) correspond to elements of a row of 𝗕 ^{p} (size 218943 × *p*).

*Case 2:*Figure 12 shows that the horizontal *T*–*T* and *T*–*S* Mexican hat covariances are well represented. Vertically (Fig. 12b), the characteristics of tracer covariances are *T*–*T* and *T*–*S* uniform within a 30-m surface mixed layer; subsurface extremum in *T*–*S* near 300 m; *T*–*T* and *T*–*S* positive down to about 1200 m where they change sign; etc. For *T*–*ψ* (Fig. 12a), quasigeostrophic balance is locally a good approximation. However, the horizontal *T*–*û* and *T*–*υ̂**π*/*f* ≃ 0.89 days, the *T*–*û* pattern at 5 m would be tilted clockwise by almost 45°. With horizontal, vertical and bottom diffusion, this tilt at 5 m is here reduced to about 30° (Fig. 12a). Note that even though inertial oscillations occur (e.g., in numerical simulations, the Ierapetra at the surface rotates in small circles at frequencies close to *f*), their effects on covariances are not always desired.^{6} In the vertical (Fig. 12b), *T*–*û* and *T*–*υ̂**T*–*ψ,* total velocity covariances near (33.80°N, 27.85°E) have a vertical minimum near 700 m. Below the zero crossings, *T*–*û* and *T*–*υ̂*

*Cases 3 and 4:* By construction, the tracer autocovariance functions of cases 3 and 2 (Fig. 12) are equal. Those of case 4 are very similar; the main difference is a volume-averaged increase in variance of 100 × (72 − 67.2)/67.2 = 7.1% (section 4b), which is distributed in the 3D *T*–*T* and *T*–*S* fields according to the tracer vectors 323 to 400. Figure 13 thus only illustrates cross-covariances with velocities. Comparing cases 2 (Fig. 12) and 3 (Fig. 13a), overall, nonlinear momentum adjustment modifies structures, but amplitudes remain similar. Focusing on differences, the horizontal *T*–*û* structure has a smaller tilt, of about 15°–20° clockwise: the nonlinear terms dissipate some inertial oscillations. Cross-covariances now extend to the Ierapetra and their gradients are tighter due to nonlinearities. The *T*–*ψ* structure has become asymmetric, with cyclonic *T*–*ψ* correlations at the Ierapetra. In the vertical, structures in the upper layers extend deeper. Other vertical changes are mainly due to the lesser 15°–20° rotation in the horizontal. Considering the velocity cross-covariances of cases 3 (Fig. 13a) and 4 (Fig. 13b), shape and amplitude differences are small. This agrees with the corresponding global similarity coefficient (Table 2b).

### c. Mesoscale: A posteriori PE fields

At the mesoscale stage (*w* = 2), the mapping procedures are as overviewed in section 4a. The differences lie in the a priori states (the subbasin-scale estimates), dataset, and mesoscale covariances (section 4b). The focus next is on cases 2 and 3 for the effects of nonlinear momentum terms, and on cases 3 and 4 for the effects of the size of the subspace.

#### 1) Tracer fields

Figure 14 illustrates the *T,* *S* components of **x**^{a} obtained by multivariate 3D analysis (appendix A section c) in case 2. By construction, the tracer fields for case 3 (not shown) are identical. Comparing cases 2 (Fig. 14) and 4 (Fig. 10 in LAL00), their tracer fields have small differences, in accord with the 4.8% (67.2%–72%) increase in the 3D tracer variance explained. Differences are largest in the surface layers and small mesoscales, which is the main variability spanned by the additional vectors 323 to 400 used in case 4. In comparison with case 1 (Figs. 9 in LAL00), results are analogous to these of section 3. The large mesoscales to the subbasin-scales have small differences, within data error bounds. The submesoscales to small mesoscales present larger differences, especially near the surface. This is because many of these small scales are environmental noise for the available data resolution (Fig. 9a) and because the 400 eigenvectors explain only 72% of the total tracer variance (section 4b).

#### 2) Transport fields

Figure 15a is the result of case 1 that computes *ψ* by vertically averaging the total velocities in thermal wind balance with the horizontal *T* and *S* analyses, assuming a level of no motion at 600 m. Figure 15b is the *ψ* component of **x**^{a} for case 2: the transport is then globally estimated from the *T* and *S* data (Fig. 9) using appendix A section c. Figure 15c is *ψ* for case 3, Fig. 15d *ψ* for case 4. The advantages of cases 2–4 are that their *ψ* estimates, and corresponding uncertainties (section 4d), are in accord with the dominant SVD of PE adjusted perturbations. Most of the regional dynamical features can be distinguished in these *ψ* fields, but not in those of case 1. Comparing Figs. 15b and 15c, the effects of nonlinear terms in the momentum adjustments agree with local mesoscale standard deviations (Table 3). Differences are largest for the Ierapetra and Mersa Matruh Gyre. Comparing Figs. 15c and 15d, increasing the size of the subspace leads to small 2% to 10% variations in *ψ,* in accord with the global covariance similarity of 86.8% (Table 2b). Largest changes are at the Ierapetra, above the Rhodes basin and along the Asia Minor Current and Mid-Mediterranean Jet.

#### 3) Internal velocity fields

Figure 16a is the result of case 1, which computes *û,* *υ̂**û,* *υ̂***x**^{a} for case 2: *û,* *υ̂**T* and *S* data (Fig. 9) using appendix A section c. Figure 16c is the same, but for case 3; Fig. 16d is for case 4. Differences between cases 1 and 2 are small and mainly due to their different tracer fields. Nonlinear momentum terms (cases 2 to 3) mainly affect the Ierapetra and Mersa Matruh Gyre. For the Ierapetra, gradient–wind balance [section 4b(3)] is a good approximation but geostrophy is not. Increasing the size of the subspace (cases 3 to 4) leads to variations in the small mesoscales, in accord with the variability spanned by the additional vectors considered. Overall, the regional increases in speed from cases 2 to 3, and from cases 3 to 4, agree with the a priori standard deviations of Table 3 and global values obtained from Table 2b.

### d. Mesoscale: A posteriori error covariance

#### 1) A posteriori mesoscale error variance

*Univariate 2D benchmark:* A posteriori tracer error variances result from the 2D analyses, for example, 5-m errors on Fig. 9b. Velocity error variances are set assuming geostrophic balance.

*Multivariate 3D scheme:*Figures 17a–b illustrate a posteriori error standard deviations for cases 2 and 3 (for case 4, see Fig. 13 of LAL00). An important result is that error properties are as one expects dynamically even though only 322 or 400 vectors are used. The tracer errors of cases 2–4 are similar to those of case 1 (Fig. 9b). Differences are due only to the use in cases 2–4 of a priori *T* and *S* covariances with nonzero multivariate and vertical correlations. By construction, tracer errors for cases 2 and 3 are identical (up to machine precision).

In general, the structures of velocity errors (Figs. 17a–b) differ from those of tracer errors, which mimic the patterns of hydrographic ship-tracks. Differences in shape are much larger for *û,* *υ̂**ψ* errors (under hydrostatic influence). The relationships among velocity error patterns, regional dynamical features, and data horizontal and vertical resolutions (unsampled regions, distances between ship-tracks, spacings between profiles along ship-tracks, nonuniform profile lengths, etc.) are discussed in section 5.b.ii of LAL00. These discussions apply here. Focusing on sensitivities, velocity errors for cases 2–4 differ from these of case 1 (not shown) because, in cases 2–4, a priori error covariances involving velocities are nonhomogeneous and anisotropic in 3D and because the dominant momentum balance is not everywhere geostrophy (diffusion and nonlinearities matter locally). Comparing cases 2 (Fig. 17a) and 3 (Fig. 17b), the nonlinear momentum terms impact a posteriori errors in *ψ,* *û,* and *υ̂*

#### 2) A posteriori mesoscale error eigenvectors and spectrum

The first and second nondimensional a posteriori error singular vectors for cases 2 and 3 are illustrated on Figs. 18 and 19, respectively (for case 4, see Fig. 14 of LAL00). For each of cases 2–4, the data **y**^{o} have redistributed the variance in the subspace. Dominant a posteriori vectors have little in common with a priori ones. In the linear case 2 (Fig. 18), the two dominant vectors are associated with the low data resolution along 30.75°E (Fig. 9a) and lack of data in the northeast corner (north of 35.5°N, east of 29.5°E). In cases 3 (Fig. 19) and 4, this holds for the first vector, but not for the second. The second vector then corresponds to low data resolution patches centered on 29°E, which is across the Mersa Matruh-Shikmona Gyres where nonlinearities matter (section 4b). In case 2, the first vector accounts for 5.7% of the normalized a posteriori error variance; the second for 5.1%. In case 3, these numbers are smaller, 5.2% and 4.8%, because some additional kinetic energy due to nonlinear terms remains unmeasured in the a posteriori eigenvalue spectrum. In case 4, these numbers further reduce to 4.25% and 4.05% due to the larger subspace size. Overall, eigenvalue spectra differ for the same reasons, but by smaller amplitudes. In case 2, the dominant 50 vectors explain 75.8% of the variance explained by all 322 vectors; the dominant 100 explain 87.2%. In case 3, these numbers remain nearly unchanged, 75.5% and 87.2%. In case 4, they are logically slightly smaller: 71.5% for 50 vectors and 84.5% for 100 vectors out of all 400 vectors. For all cases, in comparison to a priori values [section 4b(2)], the analysis has here steepened the top and flattened the middle and end of the error spectrum.

Structurewise, cases 2–4 have almost the same first vector (Figs. 18–19, panels 1a–b). Due to the nonlinear adjustments, other vectors are reordered in cases 3–4: for example, up to effects of the orthogonality constraint and arbitrary sign, the second vector of case 2 is as the third of case 3 (not shown) and inversely (Fig. 19). However, the dominant vectors of cases 3 and 4 are almost identical, indicating a convergence. Locally, both dynamics and data affect all vectors. Focusing here on data coverage (see LAL00 for dynamical effects), consider, for example, the first vector of cases 2–4 (Figs. 18–19, panels 1a–b). For each variable, amplitudes north of 35.5°N in the corner without data are at least twice as large as they are south of 35.5°N. The zero crossings (maxima) of the *T* and *S* patterns of this first vector are also aligned with the (lack of) meridional ship-transects (Fig. 9): from 34°N to 35.5°N included, zero crossings are every 0.5° latitude, while north of 35.5°N, the last lobe covers 0.8° latitude, across the no-data corner. Profile depths also influence error patterns. For example, east of 30.25°E, all profiles are about 1000 m deep, except for two at (35.5°N, 31°E) and (34.5°N, 31°E), which are 2000 m deep. Logically, these two deep profiles reduce the error on *û* at 35°N but not on *υ̂*

#### 3) A posteriori mesoscale error covariance functions

Comparing a priori [Figs. 12–13, section 4b(4)] and a posteriori (Figs. 20–21) error covariance functions also shows data impacts. All amplitudes are reduced in accord with the data coverage and measurement error model (appendix A). At (33.80°N, 27.85°E), both the horizontal and vertical correlation scales have been reduced to small mesoscale values.

*Case 2:* Horizontal structures (Fig. 20a) clearly show the influences of the low data resolution south of 34°N within 28°–29°E and of the three ship-tracks along 26°E, 26.5°E, and 27°E (Fig. 9a). In the vertical (Fig. 20b), *T*–*T* and *T*–*S* structures have stronger negative lobes and are more surface intensified than a priori (Fig. 12) in part because a priori *T* and *S* data errors are modeled to decay with depths (Fig. 23 in section 5b). The *T*–*û* and *T*–*υ̂**T*–*û* is now to the east, in accord with thermal wind effects and lower data resolution in that direction.

*Cases 3 and 4:* Only case 4 is shown (Fig. 21). Case 3 is analogous, up to amplitudes that are smaller in accord with the global similarity coefficient (Table 2b). In the horizontal (Fig. 21a), structures in case 4 are similar to these in case 2 (Fig. 20a), except that gradients are tighter due to nonlinearities and larger subspace size. In cross-covariances with velocities, differences are largest where features have been shown to be most nonlinear (e.g., Table 3). In the vertical, for all covariances, gradients are tighter than in case 2. Vertical patterns in *T*–*û* and *T*–*υ̂**T*–*T* and *T*–*S* are more complex, in accord with the additional small mesoscales explained by the vectors 323–400.

## 5. Quantitative evaluation of fields and impact on forecast performance

The two mapping schemes and three error subspaces (cases 2–4) are now evaluated by measuring the skill of forecasts initialized from the four analyses on 27 March (cases 1–4). Note that for analyses centered on other dates (not shown), results are analogous. Starting from each gridded state (e.g., Figs. 14–16), the numerical PE model is first integrated forward in time. Forecasts are then compared to in situ hydrographic data and satellite sea surface temperature (SST) data. Due to the interactions between tracers and velocities, the effects of the initial flow fields and velocity error subspaces are also assessed.

During 26 March–8 April, in situ samplings are relatively intensive (see Fig. 23a below) and the corresponding CTD profiles are used to evaluate forecasts. The high-resolution satellite images (GISIS 1995) are interpolated onto the numerical model grid, masking cloudy areas. During 27 March–3 April, on 3 days, different halves of the grid are clear and contain limited noise (e.g., Fig. 22a). This leads to a relatively good week composite, even though one should beware of nonsynoptic patching (LLA98). These interpolated day and week images are used to evaluate forecasts.

An issue is that the interval during which forecasts can be compared is limited by the predictive capability of the prediction system. Beyond this limit, nonlinear integrations of data and model errors could reverse conclusions. For the area, period, regime of interest, and data available, the present limit is near 10 days (LER97). Forecasts are thus not compared beyond that. Remaining issues are to define the error and choose its measure. Since comparing forecasts to analyses of data can be misleading, data and forecasts are here directly compared at data points. These data-forecast residuals can be a good estimate of forecast error, in so far as they are larger than the errors in the data. They are measured by root-mean-squares (rms), correlation coefficients, and pattern correlation coefficients (PCC).

### a. Satellite SST data

Considering first daily images (e.g., Fig. 22), it is challenging to determine qualitatively which of the forecasts is closest to the satellite SST. Quantitatively, their global scores are indeed similar (see Table 4 below). Focusing on differences, overall, gradients are more accurate in the multivariate 3D scheme (cases 2–4) than in the benchmark (case 1). The branch of the Mid-Mediterranean Jet near 33°N (on Fig. 22a, seen starting at 29°E) is also better represented in cases 2–4 (Figs. 22c–e). The lobes of the Mersa Matruh Gyre are visible in cases 1 and 4 (Figs. 22b,e), but not so much in the cases with the smaller subspace size (Figs. 22c,d). For the Ierapetra and its surrounding cold vortices, cases 1 and 4 have similar large mesoscales. At the small mesoscales, cases 1 and 4 are better than cases 2 and 3; overall, the Ierapetra gradients of case 4 are the most accurate. Comparable comments hold for other daily images (not shown). Considering the week composite (Fig. 30 in LLA98), similar statements can again be made, including for the Rhodes Gyre.

In Table 4, the skill of forecasts generated from the different analyses is evaluated quantitatively. For the three relatively clear days, the correlation coefficient between the satellite SST day composite and estimated 5-m temperature for that day is given (Table 4a). To obtain these coefficients, the horizontal mean of each field is removed first. In Table 4b, the week composite is compared with the week average of the day-to-day 5-m temperature forecasts. To take advantage of the larger composite coverage, two more measures are evaluated: (i) the rms of the residuals between the clear satellite data and averaged forecasts, and (ii) the PCC defined by PCC ≐ [𝗧^{S} − 𝗧^{b})^{T}(^{b})]/[ ‖ **T**^{S} − **T**^{b} ‖ _{2} ‖ **T**^{b} ‖ _{2}], where the vector 𝗧^{S} is the satellite week composite, ^{b} the background set to the satellite month composite for March 1995. In all computations, only common cloud-free points are considered. For the day snapshots (Table 4a), the main difference is on 27 March: the nowcasts of cases 2–4 are about 15% better than that of the univariate 2D case 1. On other days, a surprising result is that there is little variation among cases. This is mainly due to uncertainties in the satellite data, even though it also reflects the subspace and forecast convergences. Another artifact of data errors and also varying data coverage is the increasing correlation with time. For the week composite (Table 4b), correlation coefficients are analogous. The PCCs indicate that case 1 is the worst and case 3 the best (adding more vectors can thus at times decrease performance). These PCC numbers are relatively low. This is mainly because they focus on the small mesoscale for which the PCC skill is smaller than for the large mesoscale and subbasin scale due to the initial data resolution. The rms's indicate an error slightly larger for case 1 than for cases 2–4, but variations are not significant (the estimated error in the in situ data at the surface is around 0.3°C, see Fig. 23 below). For the day images, the rms and PCC (not given) lead to the same conclusions. Overall, the satellite data indicate that the skills of cases 2–4 are similar and only a few times slightly larger than the skill of case 1.

### b. In situ data

In the evaluations based on in situ *T* and *S* data (Fig. 23a), time is divided in periods of 3 days. For each of these periods, data-forecast residuals are computed at the intersections of the CTDs with the model grid, using the forecast for the center of the period (measurement model details are in LER97). These residuals are measured by rms, averaging level by level. On Fig. 23b, the resulting rms error (rmse) estimates are plotted for the temperature and salinity of cases 1 and 4 on 12 horizontal model levels (from 0 to 500-m depth). Overall, forecast errors increase with forecast durations and decrease with depths in proportion to the variability. For 26–28 March (Rhodes Gyre area) and 29 March–1 April (near 30°E), most rmse's are below the estimated error standard deviations of the data. Both schemes likely lead to good forecasts. Focusing where rmse's are significant, case 4 is better than case 1 at the surface, but the opposite holds near 500 m. For 3–5 April (data along the Cretan Arc), rmse's are much larger, in accord with the longer forecast duration and also lesser quality of the numerical model in this region. For *T,* case 1 is better than case 4 at all depths. For *S,* case 4 is substantially better than case 1 in the surface layers, but is worse below. For 6–8 April, at the surface, case 4 is better for *T,* but not for *S.* At middepths, the situation reverses. Near 500 m, case 1 is best but errors become smaller than the estimated data error.

These results first confirm that the two schemes have similar skill. They also restate that the univariate 2D scheme can be too sensitive to small-scale noise (e.g., section 3b), hence the usually lesser skill in the surface layers. They show that the present multivariate 3D scheme can sometimes be less effective at depth. To improve this, either the subspace can be enlarged or the specification of the initial covariance decomposition (appendix A sections a–b) refined. For example, in coastal regions, the lack of better performance at depth is likely due to the initial tracer covariances that were simplified (e.g., horizontally homogeneous, vertically dominated by open-ocean data structures instead of coastal ones).

## 6. Summary and conclusions

The foregoing study investigated the influence of a priori parameters on the error subspace estimation and mapping methodology introduced in (Lermusiaux et al. 2000, henceforth LAL00). The sensitivities of the subspace and of the a posteriori gridded fields and errors to the size of the subspace, scales considered, and nonlinearities in the dynamical adjustments were exemplified and studied for mesoscale to subbasin-scale physical fields in the northwestern Levantine during 10 February–15 March and 19 March–16 April 1995. A posteriori field and error estimates were evaluated by comparison among each other and by using the univariate 2D analysis scheme of HOPS as a benchmark. Forecasts generated from the various analyses were qualitatively and quantitatively compared to in situ and satellite data. A simple back-of-the-envelope rule for the required size of error subspaces was also outlined and utilized.

Several three-dimensional, multivariate, and multiscale properties were illustrated. The global mapping was shown to estimate simultaneously all physical fields and errors from the tracer data, usefully decompose multivariate covariances, improve vertical correlations and the filtering of environmental noise, yield more accurate field gradients, and lead to velocities that are close to being PE adjusted and a posteriori velocity errors that are PE consistent. In comparison to the geostrophic benchmark, gains were largest where topographic, nonlinear, and diffusion terms mattered. In particular, a posteriori errors were found to reflect complex properties of both the data utilized and the regional dynamics (not always close to geostrophic). Based on the forecasts from the various analyses, the multivariate 3D scheme performed as well as the benchmark. Similar conclusions have been drawn in meteorology (Andersson et al. 1998). Presently, this is likely in part due to the limited data.

Based on sensitivities to scales, for a fixed domain size, smaller scales usually require larger subspaces due to spectral redness. Based on sensitivities to the size of the subspace, truncating the a priori error covariance to a converged subspace is efficient and a priori error eigenvectors of eigenvalues negligible with respect to data errors can be neglected. In the Levantine, increasing the subspace size usually improved the analysis at depths and the small mesoscales. Overall, the variations of a posteriori fields with the parameters of the subspace were as indicated by a global similarity coefficient: if a priori error covariance matrices differed by about 5% (in the standard deviation sense), so did the fields and a posteriori errors. Based on sensitivities to the dynamics of adjustments, it was found however that even when subspaces are globally similar, some of their dominant vectors can be quite different. For example, the nonlinear momentum correction modified the dominant vectors from large-mesoscale variations of the Rhodes Gyre to mesoscale variations of the Ierapetra eddy. It also modified the shape of error spectra, the structures of covariance functions, and some properties of the a posteriori fields and errors. Relative linear–nonlinear differences were computed and as anticipated found to be largest for the features and regions with high shears, around deep eddies or gyres (e.g., Ierapetra, Mersa Matruh Gyre), along fronts (e.g., Mid-Mediterranean Jet), and in Straits. The impacts of inertial oscillations on the subspace adjustment were also discussed.

The study of a priori tracer and velocity variances, spectra of eigenvalues, shapes and amplitudes of dominant multivariate eigenvectors, and structures of (cross)-covariance functions revealed several dynamical properties. The spatial distribution of characteristic hydrographic and velocity scales as well as dominant dynamical balances were quantitatively obtained for most features of the northwestern Levantine. In particular, the Ierapetra eddy was found to be close to gradient–wind balance in the spring of 1995. Coastal-trapped waves were also inferred to be likely along the northern escarpment of the basin and to account for a significant portion of the global variations of momentum and density fields in the region.

Focusing on research directions, a first issue relates to data availability and accurate multiscale statistical models. Mainly for fair comparisons with the univariate 2D scheme but also for lack of sufficient data, a successive correction approach was used instead of a general one for multiple interacting scales. A priori tracer covariance functions were also assumed homogeneous and isotropic in the horizontal instead of being nonhomogeneous and anisotropic in 3D as velocity covariances were. These two assumptions can be removed in principle (LAL00). In regions with more data than here (to allow accurate fits), the performance of the 3D scheme may then be improved. Another direction concerns algorithms for efficient dynamical adjustment of the error subspace. For example, specific models or processes (e.g., free surface, coupled biochemical irreversibilities, stochastic internal waves) may require specific adjustment procedures. Additional convergence criteria (appendix A) should also be investigated. For example, our experience in other regions suggest that taking into account each variable separately as well as all variables as a whole [Eq. (A22)] is useful. In these criteria, ideal normalizations for specific purposes should also be researched. Most results obtained here apply to the analysis step of data assimilation schemes based on error subspaces. Other research opportunities thus include: error forecasting and adaptive sampling (e.g., Lermusiaux 1999b), model improvements (e.g., LER97), dynamical studies (e.g., Lermusiaux 2001; Lermusiaux and Robinson 2001), and interdisciplinary applications (Robinson and Lermusiaux 2001).

## Acknowledgments

I am grateful to Prof. D. G. M. Anderson, Dr. C. J. Lozano, and two anonymous referees for constructive comments on the manuscript, and to Prof. A. R. Robinson, Dr. P. J. Haley, and Mr. W. G. Leslie for their encouragements. I thank the individuals who participated in the LIW95 experiment, the POEM collaboration sponsored by the IOC of UNESCO, Mr. R. Meisner (DLR) for answering questions on the GISIS filtering of the satellite data, and Ms. M. Armstrong for preparing some figures. This work was supported in part by the Office of Naval Research under Grants N00014-95-1-0371, -97-1-0239, and -00-1-0771 to Harvard University.

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## APPENDIX A

### A Priori Error Subspace Estimation and Mapping Methodology

^{t}) state vector

**x**

^{t}∈ ℜ

^{n}is assumed subject to the stochastic dynamical and measurement models, respectively,

*η*^{t}a random process of zero mean and covariance matrix 𝗤,

**y**

^{o}∈ ℜ

^{m}the data vector,

**a random process of zero mean and covariance matrix 𝗥. The time**

*ε**t*

_{0}for the estimation is fixed and 𝗥 includes a term for data decorrelation with time (see footnote 2 and LER97). An unbiased estimate of

**x**

^{t}(

*t*

_{0}) is denoted by

**x**∈ ℜ

^{n}. The state error covariance is defined by 𝗣 ≐

**x**−

**x**

^{t}(

*t*

_{0})][

**x**−

**x**

^{t}(

*t*

_{0})]

^{T}} ∈ ℜ

^{n×n}, where

*p*is the portion of ℜ

^{n}that is spanned by the eigenvectors corresponding to the dominant

*p*eigenvalues of a (normalized) error covariance. With this rational reduction, the optimal estimate becomes

**x**

^{t}(

*t*

_{0}) is subject to (A1a)–(A1b). The superscripts (

^{b}) and (

^{a}) refer to estimates before mapping (a priori) and after mapping (a posteriori). The superscript (

^{p}) indicates a rank-

*p*approximation: for example, the a priori/a posteriori principal error covariance 𝗣

^{bp}

^{ap}

*p*eigendecomposition of 𝗣

^{b}/𝗣

^{a}. Quantities marked with asterisks are normalized.

#### Observed portions of the a priori error covariance

*ϕ*at location (

**r**,

*z*), where

**r**= (

*x,*

*y*) and

*z*are horizontal and vertical positions in a suitable coordinate system, is denoted by

*ϕ*(

**r**,

*z*). The a priori error covariance function for the fields

*ϕ*(

**r**

_{1},

*z*

_{1}) and

*ϕ*(

**r**

_{2},

*z*

_{2}) is then,

*Z*

^{*}

_{ϕi}

*z*)'s and

*Z*

^{*}

_{ϕi}

*z*)'s are vertical functions, normalized in some suitable sense. Substituting (A3a)–(A3b) into (A2) yields,

*A*

_{ϕi,ϕj}

*A*

_{ϕj,ϕi}

*ϕ*

_{i}(

**r**)

*ϕ*

_{j}(

**r**)} and

*C*

^{*}

_{ϕi,ϕj}

**r**

_{1},

**r**

_{2}) ≐

*ϕ*

_{i}(

**r**

_{1})

*ϕ*

_{j}(

**r**

_{2})}* the nondimensional horizontal cross-covariance function associated with

*ϕ*

_{i}and

*ϕ*

_{j}. The representation (A4a) is efficient if

*I*and

*J*(i.e.,

*A*

_{ϕi,ϕj}

*i*and

*j*).

In section 3c of LAL00, algorithms for estimating the dominant components of error covariance matrices based on the functionals (A4b) are outlined for three types of assumptions involving a single scale, multiple but independent scales, and general multiple scales. In each of these assumptions, the vertical and horizontal decompositions can be obtained either directly from data (EOFs of scale-restricted data residuals) or from an analytical model fit to data (eigendecomposition of a specified covariance matrix).

#### Present construction of the a priori error subspace

Here, the discrete state vector **x** ≐ (**û**, **v̂**, 𝗧, 𝗦, **p**) ∈ ℜ^{n} contains gridded values of the *û,* *υ̂**T,* *S,* and *ψ* fields (see footnote 1). The corresponding a priori error covariance estimate, 𝗕 (superscript^{b} omitted), is constructed as follows.

*T*and

*S,*the assumption of multiple but independent scales is made and (A4b) becomes

*ϕ*and

*ϕ*any four combinations of

*T*and

*S.*As in the univariate 2D scheme, for each scale

*w*in (A5),

*R*

^{*}

_{TwTw}

*R*

^{*}

_{SwSw}

*R*

^{*}

_{TwSw}

_{trc}of 𝗕 is then the sum of Kronecker products of vertical

^{z}

_{trcw}

^{r*}

_{trcw}

^{p}

_{trcw}

*w,*

^{z}

_{trcw}

^{r*}

Complete multivariate formulation: The velocity portions of 𝗕 are computed through an ensemble of adjustment momentum integrations. The algorithm for a given scale *w* is outlined below. An ensemble of perturbed tracers **x**^{bj}_{trc}^{bj}^{bj}*p* until (A22) determines that the number of samples is large enough. Once this occurs, 𝗕 ^{p} is obtained (A23). In (A22), the pairs (𝗘*, **Π** = 1/*p* **Σ**^{2}) of rank *p* and (**Π̃***p̃* ^{2}) of rank *p̃* ≤ *p* correspond to “new” and “previous” estimates of 𝗕 ^{p}. With the weighted inner products, **Π̃**^{1/2} ^{T}𝗘* **Π**^{1/2}, both principal error directions and amplitudes are compared (other criteria are in LER97). In (A23), **Γ** is a scaling, block diagonal matrix.

#### Error subspace mapping scheme

**x**

^{b}, 𝗕

^{p}(A23) and (A1a)–(A1b), the extremum of (A1c) yields the a posteriori estimates

**x**

^{a}, 𝗘

^{a}, and

**Π**

^{a}, hence 𝗕

^{ap}

^{a}

**Π**

^{a}𝗘

^{aT}

**x**

^{b}and

**x**

^{a}is hypothesized to be a linear function of

**x**

^{b}and

**y**

^{o}−

**x**

^{b}), each of which is assumed unbiased. The extremum is then (LER97),

**x**

^{b}, 𝗛

^{p}≐ 𝗛𝗘, the columns of 𝗨 are ordered orthonormal eigenvectors of

**Π̃**

^{a}, and

**Π**

^{a}is diagonal. With the successive corrections, (A24a)–(A24c) are repetitively used for each scale

*w.*The Kalman update and (A24a) differ because of the different gains, 𝗞 ≐ 𝗣

^{b}𝗛

^{T}(𝗛 𝗣

^{b}𝗛

^{T}+ 𝗥)

^{–1}and 𝗞

^{p}≐ 𝗕

^{p}𝗛

^{T}(𝗛 𝗕

^{p}𝗛

^{T}+ 𝗥)

^{–1}, respectively. Three cases are discussed in LLA98 as a function of the measurement properties 𝗛 and 𝗥.

## APPENDIX B

### Relevant Scalings and Nondimensional Numbers

Table B1 defines numbers that are relevant for the present mesoscale to subbasin-scale study. The notation is standard (Gill 1982; Pedlosky 1987; Cushman-Roisin 1994): *g* is the acceleration due to gravity; *L* and *D* the horizontal and vertical length scales of motion; *U* and *W* the horizontal and vertical characteristic velocity scales; *f*_{0} and *β*_{0} the Coriolis parameter and its northward gradient at latitude *θ*_{0}; *ρ*_{s}(*z*) the background density profile; and *ρ*_{0} a constant reference density. In addition, *N*_{s} = *R*_{D} = *N*_{s}*D*/*f*_{0} and *R*_{ext} = *f*_{0} the internal and external Rossby radius of deformations, and ∇_{h}*H* the horizontal gradient of bottom topography *H.* Except for *g,* all of these characteristic scales are here regional; they are functions of the location and process considered.

For the synoptic and horizontally isotropic quasigeostrophic scales in a stratified ocean, Ro, ^{–1}_{ext}*s* ≪ 1, but the other numbers may take a range of values: in particular, Bu and *α* increase with the strength of the stratification and topographic slope, respectively (e.g., Pedlosky 1987). At these scales, using the internal energy (density) equation to estimate *W,* one can also show that Ro/Bu is the ratio of vertical convergence to horizontal divergence *WL*/*UD,* and that the characteristic horizontal variations in density Δ*ρ* and vertical variations in background density Δ*ρ*_{s} are related via Δ*ρ*/*ρ*_{0} = (Ro/Bu)(Δ*ρ*_{s}/*ρ*_{0}).