## 1. Introduction

We evaluate the response of a general circulation model (GCM) to large-scale density perturbations in the North Pacific and construct a simple linear model that approximately describes the time evolution of these perturbations. This study is of interest both because it provides insight in the large-scale response of the model ocean and because of the possible use of the linear model for data assimilation leading to improved estimates of oceanic climate. The study was initiated in anticipation of long-range tomographic measurements to be made by the Acoustic Thermometry of Ocean Climate (ATOC) project in the North Pacific. An example is presented using simulated tomographic and altimetric observations.

A fundamental requirement for monitoring oceanic climate and climate shift is the separation of the large, slow scales of oceanic variability from the mesoscale and other short-term variability. This separation of scales is often performed using low-pass filters (Levitus et al. 1994; Parrilla et al. 1994; Roemmich 1992) or objective analysis procedures (Bindoff and Wunsch 1992; Fukumori and Wunsch 1991; Levitus 1990). But the expected amplitude of the climate signal is small relative to the natural variability, and the available observations are scarce and disparate. Our objective is to improve the estimates of oceanic climate by combining the available observations with a modern GCM.

The optimal combination of oceanic data and models is extensively discussed by Wunsch (1996), Bennett (1992), Ghil and Malanotte-Rizzoli (1991), and references therein. The most successful estimation methods, that is, the ones that can provide optimal state estimates and error statistics, are applicable to linear or linearized problems. However, most oceanographic and other real-world problems are intrinsically nonlinear.

Another complication is the computational burden of the estimation algorithms. The complete characterization of the error statistics requires the storage and manipulation of a covariance matrix with a dimension equaling the square of the model state. For example, one of the models used in the current study has a state size of 10^{6}, which is modest by modern GCM standards. This model requires 1 h of processing time per year of integration at a sustained rate of 1 Gflop s^{−1}. The error covariance matrix would have 10^{12} elements and require on the order of 100 yr of processing time per year of integration for its computation. Hence, the brute-force solution of the estimation problem is impractical for the foreseeable future, even with the anticipated availability of teraflop-per-second supercomputers.

For the above reasons, the art of data assimilation often resides in finding ways to linearize and reduce the dimensions of the problem at hand. Fukumori and Malanotte-Rizzoli (1995), and Stammer and Wunsch (1996) provide recent examples of practical estimation methods for use with large nonlinear GCMs. The former is a sequential method based on a reduced-order steady-state, linearized Kalman filter. The latter is a “whole domain” inversion based on the computation of model Green’s functions. Although the two approaches were applied to different domains and dynamical regimes, they both make use of reduced effective model dimensions and of time-invariant linearization of the dynamical model.

The current study is an extension of the work reported by Stammer and Wunsch (1996). They used a 4-level, 1° North Pacific realization of the Geophysical Fluid Dynamics Laboratory (GFDL) model and constrained it to consistency with a year of TOPEX/Poseidon altimetric data. They linearized the numerical model by computing its response to a series of isolated, geostrophically balanced vortices. The resulting Green’s functions provided the kernels for a whole domain linear inverse problem. The perturbation analysis is repeated here at higher vertical resolution, 20 levels instead of 4. In addition to the GFDL model, we also make use of a new GCM developed at the Massachusetts Institute of Technology (MIT). The Green’s functions computations are initialized using large-scale density perturbations instead of large-scale vortices. This approach excites a predominantly baroclinic response and is better suited to the study of the large, slow scales of oceanic variability. The baroclinic response is very sensitive to the details of the internal oceanic structure, while the depth-averaged response is sensitive to surface wind forcing and topography.

The model Green’s functions can be represented efficiently by a state-transition matrix instead of the explicit storage of a set of response functions. For typical problems, the former representation reduces the computational cost and storage requirements of the Green’s functions by more than an order of magnitude, and the estimation problem can be solved by a sequential filter/smoother formalism instead of a whole domain approach. The increased efficiency greatly extends the size of problems that can be tackled with currently available computational resources.

The remaining discussion is organized as follows. The climate estimation problem is formally defined in section 2. Section 3 is a brief description of the GCMs used in the current study. The response of the MIT GCM to internal density perturbations is discussed in section 4, setting the stage for the state-reduction approximation of section 5. Two methods for obtaining a linearized model are presented in section 6. In section 7 we provide a numerical example of the estimation problem using simulated altimetric and tomographic measurements, and we evaluate the sensitivity of the solution to a priori statistical assumptions.

## 2. Problem statement

**(**

*ξ**t*) represent the state of the ocean at some time

*t*and

**(**

*ξ̂**t*)

**(**

*ξ̂**t*+ 1) =

**(**

*ξ̂**t*),

**w**(

*t*)]

**w**(

*t*) represents boundary conditions and model parameters at time

*t.*We make the fundamental assumption that for large scales, the difference between the true and model states,

**x**

*t*

**B**

*ξ**t*

*(*

**ξ̂***t*)]

**x**(

*t*) are essentially linear:

**x**

*t*

*δt*

**A**

*t*

**x**

*t*

**q**

*t*

**x**(

*t*) is the reduced state vector,

**A**

*t*) is the state transition matrix,

**q**(

*t*) is the control variable that accounts for model error, and

*δt*represents the linearized model time step that in practice can be considerably longer than the time step of the GCM. We use

*δt*= 30 days for the linearized model and 1-h time steps to integrate the MIT and the GFDL models.

**B**

**B**

**B**

**B**

**I**

**BB**

**I**

*ξ**t*

**(**

*ξ̂**t*)

**B**

**x**

*t*

*ε*

*t*

**B**

**I**

*ε*(

*t*) represents the high-frequency/wavenumber components that lie in the null space of the transformation

**B**

*ε*

*t*

**O**

**O**

**x**(

*t*) are approximately dynamically decoupled from the unresolved scales

*ε*(

*t*); that is,

**B**

**(**

*ξ̂**t*) +

*ε*(

*t*),

**w**(

*t*)] ∼

**B**

**(**

*ξ̂**t*),

**w**(

*t*)]

**(**

*ξ**t*) plus noise

**(**

*ν**t*);

*η**t*

**E**

*t*

*ξ**t*

*ν**t*

**E**

**y**(

*t*) is expressed in terms of the reduced state

**x**(

*t*). The noise term

**n**(

*t*) now includes a term due to small-scale, high-frequency variability

**(**

*ε**t*) in the null space of

**B**

**(**

*ν**t*) from (8);

**n**

*t*

**E**

*t*

*ε**t*

*ν**t*

**(**

*ν**t*) is often negligible relative to the sampling error

**E**

*t*)

**(**

*ε**t*) in (11). It should be pointed out that in an analogous manner,

**(**

*ν**t*) contains a contribution due to the unresolved scales and missing physics of the GCM.

The problem consists in solving for **x̂**(*t*), the reduced state estimate, and its uncertainty, **P** = 〈(**x̂** − **x**)(**x̂** − **x**)^{T}〉, given measurements **y**(*t*) and a priori covariance matrices **Q****qq**^{T}〉**R****nn**^{T}〉**S****xx**^{T}〉**A****B****E****Q****R****S****A****B****Q****R****S**

## 3. Model description

The current study was initiated using the GFDL numerical code and model output from a global eddy-resolving integration by Semtner and Chervin (1992). These results are reported in sections 6 and 7. We have now switched over to the newly developed MIT GCM. This model is used to carry out the perturbation analysis reported in section 4 and will be the focus of our future assimilation efforts. The above models and their configurations are briefly described below.

### a. MIT model

In its current configuration, the MIT GCM (Marshall et al. 1997a,b) solves the incompressible Navier–Stokes equations in spherical geometry, has a rigid lid, and employs an equation of state appropriate to sea water. Height is used as a vertical coordinate, and the model can handle arbitrarily complex coastlines, islands, and bathymetry. The model relaxes the hydrostatic approximation but retains a “hydrostatic switch” that, if desired, turns off nonhydrostatic terms for use in large-scale modeling. It is prognostic in three components of velocity, temperature, and salinity and diagnostic in pressure. A finite volume, predictor–corrector numerical procedure is used on a staggered (Arakawa “C”) grid. The model is implemented on parallel machines.

For this study, the MIT GCM is integrated in hydrostatic mode for the Pacific Ocean. It has realistic coastlines and bottom topography (Fig. 2). Bottom and side walls are insulating. A no-slip side wall condition is imposed and the bottom is free slip. The model domain extends from 30°S to 61°N, meridionally, and from 123° to 292°E, zonally, with horizontal grid spacing of 1°. There are 20 vertical levels (see Table 2), to a maximum depth of 5302 m. At the surface, the model is relaxed to climatological values of temperature and salinity with a relaxation timescale of 25 days. At the southern boundary, the relaxation occurs over a 500-km zone with a timescale of 5 days at the boundary decreasing linearly to 100 days at 500 km.

The model was initialized from climatological annual mean temperature and salinity distributions (Levitus 1982), and a resting flow field. It was integrated for 17 years with annual mean temperature, salinity, and surface wind forcing. From year 18 onward, monthly temperatures and seasonal salinities were used, and the surface was forced with the monthly winds provided by Trenberth et al. (1989). The forcing fields were updated every 24 h using linear interpolation from the monthly or seasonal values. Surface heat and freshwater fluxes from Oberhuber (1988) were introduced in the surface layer starting on year 29, while continuing to relax to climatological temperature and salinity. The model time step is 1 h. Table 3 lists the mixing and diffusion coefficients.

Figure 3 displays the pressure and velocity fields at the 38-m depth, after 43 years of integration. The major climatological circulation components of the North Pacific can be recognized (compare to Pickard and Emery 1990, Fig. 7.31). They include the Kuroshio and the Oyashio Currents, which join to give rise to the North Pacific Current, the California Current, and the North Equatorial Current. These current systems are part of the North Pacific Gyre, and to the north, the Alaskan Gyre. The model also reproduces the South Equatorial Current, which straddles the equator, the Equatorial Countercurrent centered at 7°N, and just below the surface the Equatorial Undercurrent (see Fig. 4). Figure 5 displays temperature, salinity, and density profiles averaged over the model domain. There is a net influx of heat at the boundaries and the model gets progressively warmer, with a maximum warming of 2°C at 300 m after 43 years of integration: tracer profile gradients are severely eroded. The conclusion is that the model adequately reproduces the large-scale wind-driven circulation but fails to properly represent the small-scale processes responsible for interior property distributions. Work is under way to address these gross model errors by improving the parameterization of mixed layer dynamics and mixing processes in the interior. Nevertheless, the model as it stands suffices for the current numerical study.

### b. Bryan–Cox model

The linear model discussed in section 6 was obtained using the GFDL GCM (Bryan 1969; Cox 1984) in the configuration described by Stammer and Wunsch (1996). The model has 1° horizontal grid spacing and four vertical levels of thickness 100, 500, 1000, and 2400 m, respectively, to a maximum depth of 4000 m. It was spun up for 23 years in the Pacific Ocean north of 30°S. As discussed by Stammer and Wunsch (1996) this model simulated the major components of the North Pacific circulation described earlier. However, the model does not resemble observations in the Tropics due to the low vertical resolution

The state estimation example discussed in section 7 was carried out using output from a global implementation of the Bryan–Cox primitive equation model with nominal grid spacing of ¼° in the horizontal and 20 levels in the vertical (Semtner and Chervin 1992).

## 4. Perturbation analysis

The response of the MIT model to internal midlatitude temperature perturbations is evaluated. This problem is related to that of forced oceanic waves and geostrophic adjustment, which is extensively discussed in the literature (see Gill 1982; Philander 1978; Blumen 1972; and references therein). The difference is that the GCM provides solutions that include the effect of realistic topography, stratification, horizontal gradients, and mean circulation. The objective is to obtain guidelines for the state reduction approximation and model linearization to be discussed in subsequent sections. In particular, the extent to which the large, slow scales of the model response are dynamically coupled to the smaller or faster features resolved by the model needs to be determined.

Following the 43-yr spinup described in section 3, the MIT GCM was integrated for an additional 2 yr with monthly forcing. This 2-yr integration provides the reference state for the perturbation analysis. Starting at the beginning of this period, temperature anomalies are introduced in the model. The model is then integrated with the same boundary conditions and model parameters as before and the time evolution of the perturbations relative to the reference state is recorded.

The first experiment studies the model response to the warming of a lens of water centered at 34.5°N, 209.5°E at a depth of 350 m. The perturbation has horizontal extent of 16° (∼1600 km), both zonally and meridionally, and a vertical extent of 500 m. It is tapered by a Hanning (cosine) window in all three directions with a maximum perturbation of 0.1°C at the center. No attempt was made to introduce the corresponding geostrophic velocity perturbation because, as is shown below, the geostrophic adjustment transients can be neglected.

*g*= 9.8 m s

^{−2}is the acceleration due to gravity;

*δz*= 500 m and

*δx*= 1600 km are the vertical and horizontal length scales of the disturbance, respectively;

*f*= 8 × 10

^{−5}s

^{−1}is the Coriolis parameter; and

*δρ*/

*ρ*= 1.7 × 10

^{−5}is the fractional density change corresponding to a 0.1°C warming at that depth. The underlying cyclonic perturbation is such that the depth-integrated velocity is approximately zero so that the velocity perturbation below the temperature anomaly is

*δv*

_{b}

*h*

_{1}

*δυ*

_{s}

*h*

_{2}

^{−1}

*h*

_{1}= 350 m and

*h*

_{2}= 4650 m represent the height of the water column above and below the disturbance, respectively. In the model, the depth-integrated perturbation velocity is nonzero because of residual terms from the geostrophic adjustment process (∼10

^{−6}cm s

^{−1}) and because of nonlinear flow interaction with bottom topography, horizontal density gradients, and the mean circulation (∼0.01 cm s

^{−1}). The contribution from the geostrophic adjustment process is negligible. However, the topographic and other nonlinear effects are of the same order of magnitude as the cyclonic flow below the perturbation and they cannot be neglected. Its spatial structure would be difficult to predict from theoretical considerations alone, but it is readily provided by the numerical model.

In an analogous manner to geostrophic adjustment, but on much longer timescales, the model ocean returns to local Sverdrup balance by radiating Rossby waves away from the perturbed region (e.g., Gill 1982, section 12.4). Figures 7a and 8a show the model response at the end of month 16. The pattern is characteristic of long westward propagating baroclinic Rossby waves (see Gill 1982, section 12.3). The waves exhibit the expected decrease in phase speed with increasing latitude (the southwest-northeast slope of the patterns), but are modified by complex interactions with topography, horizontal density gradients, and the mean flow field (note the southeast drift of the pattern relative to the initial location of the disturbance). The observed phase speed of the first baroclinic mode is *c* ∼ 3 cm s^{−1} at 34.5°N, which suggests an internal radius of deformation *L*_{D} ∼ (*c*/*β*)^{1/2} ∼ 40 km, where *β* = 1.9 × 10^{−11} m^{−1} s^{−1} is the variation of the Coriolis parameter with latitude. Higher baroclinic modes have progressively slower phase speeds, causing the separation of the disturbance into several baroclinic modes. Finally, we note that the contribution of salinity to the density perturbation is not insignificant. For example, a salinity change of order 0.005 psu is observed near the surface at the end of month 16, compared to a temperature perturbation of order 0.05°C. These correspond to density perturbations of order 0.0035 and 0.01 kg m^{−3}, respectively. On average, the net effect of changes in model salinity is to decrease the total density perturbation due to the temperature anomaly.

*λ*> 1000 km [Gill 1982, Eq. (12.3.9)] where

*k*

_{h}= 2

*π*/

*λ*is the horizontal wavenumber.

In a third numerical experiment, the model was perturbed with a 0.1°C temperature anomaly in a single grid box at 34.5°N, 209.5°E and 350-m depth. This impulse function excites all spatial and temporal scales that can be resolved by the GCM. The response at the end of month 16 is shown in Fig. 7c. Although there are differences in the details, the large-scale response away from the Kuroshio region is clearly similar to that of previous experiments. This suggests that away from the western boundary, and for the current model configuration, the large-scale low-frequency response is effectively decoupled from smaller and faster processes. In the context of the current study, the complicated response near the western boundary can be accounted for by increasing the a priori error variance of the linear model in that region.

The particular response of the model ocean to perturbations is a sensitive function of the internal structure. For illustration, Figs. 8b,c display the response of an unforced ocean to the same perturbation as in Fig. 8a. The large differences between the realistic (Fig. 8a) and the idealized (Figs. 8b,c) cases indicate the rich information content of the response, as captured by a modern GCM. The estimation recipe to be discussed later seeks to use this information to better resolve the internal oceanic structure from a necessarily incomplete set of measurements.

## 5. State reduction

The total state dimension of the MIT GCM in the aforementioned configuration is 1335852. As discussed earlier, the complete solution of the estimation problem would overwhelm the most powerful computers because of the need to store and manipulate the error covariance matrix. To proceed, one can have recourse to methods where the error covariance matrix is either ignored (e.g., nudging, Malanotte-Rizzoli and Young 1992) or not absolutely necessary (e.g., the adjoint method, Marotzke and Wunsch 1993). The alternative, which is discussed here, is to use optimal estimation methods on a reduced problem.

The motivation for the current work is the estimation of oceanic climate, that is, the large or slow scales of the variability, augmented where possible by a statistical description of the smaller scales. The exact location and time history of each geostrophic eddy is not required or obtainable from existing global datasets. However, it is assumed that a GCM driven by observed meteorological fields and constrained to consistency with the available data can accurately simulate the large or slow scales without having to exactly reproduce the small-scale behavior of the ocean.

**B**

**B**

**B**

**B**

^{T}

**B**

**B**

^{T}

^{−1}

**B**

**B**

^{T}will be invertible for any but the most unfortunate choice of

**B**

**B**

**B**

**B**

**B**

**B**

^{T}

**B**

^{−1}

**B**

^{T}

**B**

^{T}

**B**

**B**

**B**

**B**

^{∗}

_{h}

**B**

^{∗}

_{υ}

**B**

^{∗}

_{t}

**B**

^{∗}

_{t}

**B**

^{∗}

_{υ}

**B**

^{∗}

_{h}

**B**

_{t},

**B**

_{υ}, and

**B**

_{h}can be defined as in (15), and it follows that

**B**

**B**

_{t}

**B**

_{υ}

**B**

_{h}

**B**

**B**

In this study, the vertical state reduction operator **B**^{∗}_{υ}**B**^{∗}_{υ}**B**^{∗}_{h}**B**^{∗}_{t}**B**^{∗}_{h}**B**_{h} can be efficiently implemented using FFTs without explicit evaluation or storage of either matrix. No time filtering is required because the fields used in the numerical examples have sufficiently red frequency spectra.

Alternative representations of the perturbation field are available. Empirical orthogonal functions (EOFs) from hydrographic casts are the preferred description for vertical structure in ocean acoustic tomography. Wavelets or climate model EOFs can be used horizontally. Temporal filtering may be required if the processes under study have frequency spectra that are not sufficiently red to avoid aliasing. A different approach consists in first defining the interpolation operator **B****B**

In their analysis of the large-scale North Pacific circulation using satellite altimetry, Stammer and Wunsch (1996) form averages in 10° × 10° areas during 10-day periods. The 10-day interval coincides with the repeat cycle of the TOPEX/Poseidon altimeter. The vertical sampling scheme consists of depth averages at the four levels of their particular implementation of the GFDL model: 0–100, 100–600, 600–1600, and 1600–4000 m. This sampling scheme resolves variability due to the propagation of large-scale baroclinic Rossby waves, but it is unsuitable to the observation of barotropic Rossby waves at midlatitudes. Long barotropic Rossby waves have short periods. For example, at wavelengths greater than 1000 km, the midlatitude period is typically less than 23 days (Pond and Pickard 1983, section 12.10.4).

The current state reduction scheme restricts the analysis to scales larger than 16° in the horizontal and periods longer than 60 days, which is sufficient to resolve the large-scale baroclinic response of the ocean discussed in section 4. However, the fast or short variability caused by the propagation of barotropic Rossby waves is eliminated. It is appropriate, therefore, to restrict the reduced state vector to the description of density perturbations. Here, temperature is used as a proxy for density, but the salt contribution can be recovered from representative *T*–*S* diagrams or directly from the model Green’s functions.

## 6. Linearization

Two methods for deriving the state transition matrix, **A****x**(*t*). These perturbations are introduced in the numerical model as **B**_{υ}**B**_{h}**x**(*t*). The response of the GCM is computed as in section 4 for a single time step, *δt* equals 1 month, of the linear model. The GCM response is then projected back onto the reduced state vector using **B****x**(*t*). This computation, repeated for each element of **x**(*t*), provides the complete state transition matrix.

For illustration, the above method is used to obtain a state transition matrix for the North Pacific GFDL integration of Stammer and Wunsch (1996) (see section 3b for a brief description of model configuration). Three hundred thirty-six 30-day Green’s functions are computed at the locations marked by the dots in Fig. 9 and used to define the state-transition matrix **A****A**

**A**

**B**

**x**(

*t*), which is consistent with GCM perturbation dynamics. Right multiplication of (3) by

**x**

^{T}(

*t*) and taking expectations yields the state transition matrix

**A**

**x**

*t*

*δt*

**x**

^{T}

*t*

**x**

*t*

**x**

^{T}

*t*

^{−1}

**x**(

*t*) and

**q**(

*t*) are uncorrelated. This method has the advantage of being able to provide an average linear model for the entire period under study rather than an exact model for a particular month.

**x**(

*t*)

**x**

^{T}(

*t*)〉 is not invertible—for example, when the number of time steps available for extracting the linear model is smaller than the dimension of

**x**(

*t*)—the following inverse problem, can be solved for the coefficients of

**A**

## 7. State estimation example

In this final section we present a numerical example of the ocean climate estimation problem making use of the tools and ideas discussed above and providing an opportunity to study the sensitivity of the solution to a priori statistical assumptions. We consider a rectangular piece of ocean in the North Pacific (10°–60°N, 140°–240°E). A 4-yr integration of the Semtner and Chervin (1992) global ocean circulation model with nominal grid spacing of ¼° in the horizontal and 20 levels in the vertical is assumed to represent the real ocean. Pseudotomographic and altimetric measurements are then inverted to recover large-scale temperature perturbations about the mean model state. Without loss of generality, a time-invariant system is considered in which **A****E****Q****R****S****q**(*t*), **n**(*t*), and **x**(*t*) are assumed uncorrelated with each other and with zero mean; **q**(*t*) and **n**(*t*) have zero temporal autocorrelation. The above assumptions are a reasonable starting point, given the coarse spatial and temporal resolution of the reduced state vector. Nevertheless, these assumptions must be checked for statistical consistency with the solution.

The basic model state, ** ξ̂**(

*t*)

**x**(

*t*), consists of 30-day snapshots of temperature perturbations about this 4-yr mean, filtered by operator

**B**

^{∗}

_{h}

**B**

^{∗}

_{υ}

**x**(

*t*).

### a. Ocean acoustic tomography

Ocean acoustic tomography (Munk and Wunsch 1979) provides time series of travel times that are inversely proportional to the integrated along-path sound speed or, equivalently, temperature. Each transmitter–receiver pair can resolve several ray or modal arrivals that have different vertical sampling characteristics. Therefore, the inversion of these arrival times provides some degree of vertical resolution along each path.

In this study the pseudotomographic measurements, ** η**(

*t*) in (8), represent the mean temperature perturbation along each of the 13 proposed North Pacific ATOC paths (Fig. 13) at four vertical levels: 0–100, 100–600, 600–1600, and 1600–4000 m. One may think of these as the result of vertical-slice inversions along each path. Figure 14 displays the rms signal

**EB**

**x**(

*t*) and the noise

**n**(

*t*) components of the observed difference

**y**(

*t*) in (10). The simulated measurements have the highest signal-to-noise-ratio in the surface layer, 0–100 m, due to the seasonal heating and cooling of the mixed layer in the model ocean. In practice, we expect poor resolution at the surface (due to the acoustics) and a stronger signal in the lower layers (larger deviation between modeled and true state of the ocean).

Figure 15 shows a contour diagram of a row of matrix **EB**** ξ**(

*t*) by

**x**(

*t*). The noise term

**n**(

*t*) is the difference between this weighted average and the mean temperature along the tomographic line, that is,

**E**

*t*)

**(**

*ε**t*) in (11).

### b. Static inversion

A singular value decomposition (e.g., Wunsch 1996) **EB****UΛV**^{T} is used to understand the nature of the solution and its relationship to the data. Here, **U****y**(*t*), **V****x**(*t*), and **Λ****EB****V**

The singular value decomposition is also a convenient way to obtain the natural solution of the estimation problem, the solution that assumes infinite a priori variance in the range space of the solution and zero variance in the null space. This solution is appropriate when no a priori statistical information is available about the model parameters and the measurements. The corresponding model resolution matrix is **V**_{p}**V**^{T}_{p}**V**_{p} is a matrix containing the first 52 columns of **V****V**_{p}**V**^{T}_{p}

*J*

**n**

^{T}

**R**

^{−1}

**n**

**x**

^{T}

**S**

^{−1}

**x**

**y**(

*t*) in (10) subject to constraint (21) using the Gauss–Markov estimators for

**x**and

**P**

**R**

**S**

**x**〉 = 0.

Figure 17 displays the explained standard deviation at each of the four model levels for a particular inversion where **S****R****x** and **n** (Figs. 12 and 14, respectively) but with no information about spatial correlations (that is, with zero off-diagonal elements). This result can be compared to the predicted deviation based on solution uncertainty **P****P**

The results of Figs. 17 and 18, as well as the results from several other inversions using different a priori statistical assumptions, are summarized in Table 4. A single figure of merit is formed at each level by averaging the explained standard deviation. Table 4 indicates that the skill of the inversions is particularly sensitive to the full and correct specification of the solution covariance matrix **S****S****S****S****R****S****R****S****R****S****x**(*t*) is red. In practice, the a priori statistics can be provided by a recent climatology of the region under study or by a realistic GCM, but we do not expect the skill of the inversions to ever approach that indicated by the bottom two rows of Table 4. However, with enough data, adaptive filter/smoother methods could be used to determine the model and data covariances.

### c. Time-dependent inversion

The sensitivity of the filter/smoother to a priori statistical assumptions was tested for several combinations of **R****Q****S****R****Q**

**R**

**S**

**Q**

**S**

**Q**

**R**

**S**

**x**(

*t*), as compared to the results reported on Tables 4 and 5. Unfortunately, for wrong a priori

**S**

**P**

### d. Satellite altimetry

**x**(

*t*) can be established. The particular case of satellite altimetry is discussed in this section. Assuming that the large-scale perturbations of the ocean relative to model predictions are in geostrophic and hydrostatic balance, then the sea surface elevation can be decomposed into a steric component plus an integration constant. The integration constant is proportional to the horizontal gradient of the depth-averaged velocity, and to first order it is not seen by one-way tomography. The steric component is the full-water column integral of density perturbations,

*ρ*(

*x, y, z, t*), producing a sea surface elevation where

*H*(

*x, y*) is the depth of the water column and

*ρ*

_{0}(

*x, y, t*) is the seawater density at the surface. For example, a 0.5°C warming of a layer between 100- and 600-m depth with mean salinity 34.7 psu and temperature 9.6°C results in a 4.3-cm sea surface elevation (see Table 7). The current definition of the state reduction operator

**B**

**E**

*t*)

**(**

*ε**t*) in (11), because they cannot be resolved by the reduced state vector.

Pseudoaltimetric measurements were constructed in the Semtner and Chervin ocean using the values of *dh*/*dT* from Table 7 and 1-cm rms Gaussian residuals, typical of TOPEX/Poseidon sea level anomaly error for the large spatial and temporal scales considered here. The results of some static and time-dependent inversion experiments making use of these pseudoaltimetric measurements are summarized in Table 8. The results of these idealized experiments indicate that overall altimetry is the richest and most useful signal, even for the deep levels. This is not surprising given the limited number of tomographic integrals as compared to the complete spatial coverage of the altimeter. Nevertheless, the combination of altimetric and tomographic measurements improves the overall skill of the inversions relative to that of either measurement on its own. It should also be pointed out that the assumption of white noise for the altimeter measurements may be unrealistic and that long-term climate-scale changes are unlikely to have the linear physics assumed here. Tomographic integrals measure temperature directly and can therefore help to ground-truth altimetry, to diagnose ageostrophic adjustments, and to separate the measured sea surface elevation into barotropic and steric components. For the reasons discussed earlier, the filter/smoother with synthetic measurements outperforms the other two types of inversions but underestimates the uncertainty of the estimates. More realistic a priori covariance matrices, instead of the diagonal matrices used for constructing Table 8, improve the results substantially.

## 8. Concluding remarks

The principal contributions of this study are the evaluation of the MIT GCM response to large-scale internal density perturbations in a North Pacific test basin, and the recipe for obtaining and using a reduced-order linearized model to estimate ocean climate. At midlatitudes, the model response is that of linear baroclinic Rossby waves, modified by complex interactions with horizontal density gradients, mean advection velocity, and, to a lesser extent, topography (see Figs. 7 and 8). Except near the western boundary, the model response to a localized perturbation is essentially similar on the large scales to that resulting from a large-scale initial perturbation. This result indicates that the large-scale model response is largely decoupled from the smaller and faster model physics.

Two methods for obtaining a reduced-order linear model that describe the evolution of the large-scale internal density perturbations have been described. The first method is based on the computation of model Green’s functions and their representation in state-transition matrix form. The second is a form of principal oscillation pattern analysis. The reduced-order linear model compares favorably with the large-scale response of the fully nonlinear GCM for a period of up to 2 years (see Figs. 9 and 10). The linear model is suitable for climate estimation studies. Numerical examples of both static and time-dependent inversions have been presented using simulated tomographic and altimetric measurements. As a result of insufficient data constraints, the inversions are extremely sensitive to the quality of the a priori statistical assumptions.

The following algorithmic improvements are suggested. For large-scale, low-frequency estimates of oceanic circulation, model Green’s functions can be initialized using internal density perturbations, instead of vortices as in Stammer and Wunsch (1996). The former excite a predominantly baroclinic response, have much longer persistence, and are easier to implement numerically. Because most of the energy of long planetary waves is in potential form, the initial geostrophic adjustment transients can be neglected. By contrast, the adjustment process cannot be neglected when the perturbations are initialized from vortices. This adjustment adds to the computational burden of the model Green’s functions as it may take up to a full month for the geostrophic adjustment transients to die down. Most of the useful information about the interior of the ocean is contained in the baroclinic response (see Fig. 8). The depth-integrated response, which is not excited by the method proposed here, is predominantly a function of bottom topography and surface wind forcing.

The state-transition matrix provides an efficient representation of model Green’s functions. This representation substantially reduces processing and storage costs (see the appendix). It naturally lends itself to the use of sequential estimation algorithms. Therefore, much larger inverse problems can be tackled with the same computational resources. Given the slow propagation of baroclinic information in the ocean, it is suggested that the response to several perturbations can be computed simultaneously, thus further reducing processing requirements. Equation (19) or (20) is then used to obtain the state-transition matrix.

Due to insufficient data constraints and imperfect knowledge of a priori statistics, the estimates provided by the time-dependent inversion can diverge from the desired solution. A simple modification of the problem formulation, Eqs. (25)–(27), can be used to require the solution to remain within bounds provided by a recent climatology or otherwise.

Important issues that have not been addressed in this study include the optimal choice of a priori covariance matrices for the oceanic state vector, measurement residuals, and system errors. Our priorities for future work are the application of the above ideas to real data, and the testing of reduced-order linear models in other basins and at higher GCM resolutions.

## Acknowledgments

We gratefully acknowledge the help of the MIT GCM team: Alistair Adcroft, Curtis Heisey, Chris Hill, John Marshall, and Lev Perelman. This study benefited from discussions with our colleagues Ichiro Fukumori, Jochem Marotzke, and Detlef Stammer. We thank Bert Semtner and Bob Chervin for making their model output available. Financial support was provided by SERDP/DARPA as part of the ATOC project (University of California SIO Contract PO#10037358) and by NASA Grant NAGW 1048. This work was also supported by grants of HPC time from Project SCOUT at MIT-LCS (DARPA Contract MDA972-92-J-1032), from the Arctic Region Supercomputing Center, and from the Scientific Computing Division of the National Center for Atmospheric Research.

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## APPENDIX

### Efficient Representation of Green’s Functions

*G*

_{ij}(

*t*) is defined as the time-evolving model response at location

*i*and time

*t*caused by a unit perturbation at location

*j*and time 0. Subscripts

*i*and

*j*can also refer to some truncated basis set, as discussed in section 5, and

*t*is the corresponding time index. If the perturbation physics of the model are approximately linear for the resolved scales, then by definition where

*N*is the state dimension. The state transition matrix is defined as Equations (A1) and (A2) yield where

*G*

_{ij}(0) = 1 for

*i*=

*j,*and 0 otherwise. Therefore Eq. (14) in Stammer and Wunsch (1996) is seen to be equivalent to (24) in this paper. This representation of the Green’s functions affords more than an order of magnitude decrease in the computational cost and storage requirements. The model response to each perturbation needs be computed for a single time step, rather than for the complete duration of the experiment. The response at future time steps can then be constructed using (A3) as shown on Figs. 9 and 10. Furthermore, the inverse problem can now be solved sequentially using filter/smoother algorithms as was done in section 7.

Specifically, the computation of the state-transition matrix requires integrating the GCM for a total of *N* linear model time steps and the storage requirement for **A***N*^{2}. The computation of the complete set of model Green’s functions requires integrating the GCM for a period of ^{M}_{i}*K*_{i}*t*_{f} and the storage requirement is ^{M}_{i}*K*_{i}*Nt*_{f}/*δt,* where *M* is the total number of measurements, *t*_{f} is the duration of the experiment, and *δt* is the linear model time step. Here, *K*_{i} represents the number of degrees of freedom assigned to measurement *i,* that is, the number of nonzero elements for the particular row of **EB**

For the example of section 7d, *N* = 336, ^{M}_{i}*K*_{i} ∼ 800, *t*_{f} = 46 months, and *δt* = 1 month. Therefore the storage requirement for the state transition matrix is *N*^{2} ∼ 10^{5} as opposed to ^{M}_{i}*K*_{i}*Nt*_{f}/*δt* ∼ 10^{7} for the complete set of Green’s functions. The computation of the state transition matrix requires *Nδt* = 28 GCM years as opposed to ^{M}_{i}*K*_{i}*t*_{f} = 3066 GCM years for the complete set of Green’s functions.

Description and definition of estimation matrices.

MIT GCM vertical levels definition for the 20-layer North Pacific integration.

MIT GCM mixing and diffusion coefficients for the North Pacific integration.

Actual and predicted explained standard deviation for static inversions using pseudotomographic measurements. Each number represents the mean explained deviation in percent at a particular level. A priori statistics are obtained here from the known **x** and **n**, but off-diagonal information is withheld in some of the examples as indicated. The second row of the table (diagonal **R**, **S****R****S**

Actual and predicted explained standard deviation for filter/smoother inversions using pseudotomographic measurements. These results can be compared with those for the static inversions reported in Table 4. The skill of the time-dependent estimates is worse than that of the static inversions when incomplete a priori **Q****R**

Actual and predicted explained standard deviation for filter/smoother inversions using pseudotomographic measurements and constrained to consistency with **S****S**

Typical values for sea surface elevation due to large-scale temperature perturbations in hydrostatic and geostrophic balance.

Actual and predicted explained standard deviation for static and time-dependent inversions using pseudoaltimetric and pseudotomographic measurements. Diagonal a priori **Q**, **R**,**S**