## 1. Introduction

Recently renewed interest in Indian Ocean (IO) observational and modeling studies is motivated by a number of factors such as the 1997–98 Indian Ocean dipole/zonal mode (IODZM; Saji et al. 1999; Webster et al. 1999; Murtugudde and Busalacchi 1999) and the fact that the IO is the only tropical ocean yet to be equipped with a moored observational array. The variability of the tropical Pacific and Indian Oceans is intimately connected through an oceanic bridge provided by the Indonesian Throughflow (ITF; see Godfrey 1996 for a review) and an atmospheric bridge through the Walker cell (see Webster and Yang 1992). Some recent studies even suggest that the IO can influence an evolving El Niño–Southern Oscillation (ENSO) in the Pacific (Wu and Kirtman 2004; Annamalai et al. 2005) and the secular trends in the IO are purported to have significant impacts on Northern Hemisphere climate variability (Hoerling et al. 2004; Deser et al. 2004). However, while our knowledge of the tropical variability of the Pacific Ocean has increased since the Tropical Ocean Global Atmosphere (TOGA) decade, and the implementation of the Tropical Atmosphere Ocean (TAO) Array (McPhaden et al. 1998), our knowledge of the variability in the IO is still limited because of the lack of sufficient observations. Under the auspices of the Climate Variability and Predictability (CLIVAR) Project, the IO Panel has proposed a 35-mooring array designed to observe the large-scale dynamical variability in the tropical IO (information online at http://www.clivar.org). Such an array is expected to complement remote sensing observations that, even though providing global, high-resolution coverage of the World Ocean, only supply information about the near-surface oceans (Busalacchi 1997). While the IO array is still in its design phase, the present planned version spans the region from 55° to 95°E and varies in latitude between 16°S and 8°N. Its distribution has been chosen subjectively based on the knowledge of the geographical location of the most energetic signals within the IO, and a set of phenomenological features such as the intertropical front (ITF), cross-equatorial overturning cell, and the location of upwelling zones. One of the interesting features of the seasonally reversing monsoons is the thermocline ridging along ∼10°S, which has many interesting climatic and biogeochemical consequences (Reverdin et al. 1986; Murtugudde et al. 1999; Xie et al. 2002). The air–sea interactions in the Arabian Sea and the Bay of Bengal are expected to enhance predictive understanding of the Asian monsoons (see Annamalai and Murtugudde 2004 for a review) and the dynamic and thermodynamic variability in the near-equatorial region is part of the IODZM and the Walker cell interactions (Hastenrath et al. 1993; Murtugudde et al. 2000). The intraseasonal variability is of special interest in the Bay of Bengal (Sengupta and Ravichandran 2001). The onset and growth phases of the IODZM are centered off of Java and Sumatra in the southeastern IO, justifying the choice of moorings in those regions (Annamalai et al. 2003; Zhong et al. 2005).

Several efforts are now under way to assist in the design of the IO array with observing system simulation experiments [OSSEs; e.g., see Schiller et al. (2004); and, in this issue, Vecchi and Harrison (2007) and Oke and Schiller (2007)]. Some of these studies scrutinize the design of the IO array as a component of a multi-instrument global observing system (as, e.g., remote sensing, Argo floats, and/or XBT drops from ships of opportunity). Here the design of the IO array is considered on its own as the moored array should provide, independently of other additional instruments, a stream of geographically constant, temporally homogeneous flow of surface and subsurface data for the study of the climate variability of the IO.

The goal of the research carried out here is twofold. First, we objectively validate the predetermined locations of the moorings through the analysis of the error structure obtained from a reduced-space Kalman filter. Second, we identify the most redundant moorings of the proposed array to find out if the proposed array may be simplified. Because a single array cannot encompass all relevant spatial and temporal scales, the focus of this study will be on determining the optimal mooring sites that best observe the large-scale, interseasonal-to-interannual variability in the IO.

The experiments used in this work are based on a set of OSSEs that allow the assessment of the impact of a new observing system in a data assimilation experiment using simulated observations. The approach used in this work differs from previous approaches (see, e.g., Halem et al. 1982; Atlas 1997; Vecchi and Harrison 2007; Oke and Schiller 2007) in the sense that the observations used in our assimilation experiments come from observed fields of satellite sea surface height (Lagerloef et al. 1999) and sea surface temperature (Reynolds and Smith 1994). Although the data to be assimilated are based on actual observations, they will be used to simulate the values of dynamic heights and upper-ocean temperatures retrieved by the moored platforms of the observation array. Using analyzed fields to simulate observations allows us to avoid the problem of how realistic the data generated by the model are. [A note aside, it might be argued that fitting the IO array to Ocean Topography Experiment (TOPEX)/Poseidon and Jason (TPJ) and Reynolds data does not provide a new source of information as SSH and SST are already well observed. Nevertheless, the costly maintenance of satellite platforms, the impact of cloud contamination of SST retrievals, and the need to further reduce errors from SST and SSH for prediction models cannot be overstated.]

To provide an objective justification of the spatial sampling “subjectively” proposed by the IO Panel, we will closely follow the method that Hackert et al. (1998) used to determine the optimal deployment of moored platforms in the tropical Atlantic Ocean of the Pilot Research Moored Array in the Tropical Atlantic (PIRATA; Servain et al. 1998). The underlying assumption in this methodology is that the placement of moorings that best reproduces the error structure of the data assimilation system is considered to be optimal. As described in section 4a, every grid point in the error field will be considered a feasible member of the observing network. The objective selection of the optimal locations is estimated with the help of a full Kalman filter (KF; Kalman and Bucy 1960) defined on a coarser grid. To address the issue of redundancy and array simplification, we will identify the most redundant moorings of the array. Redundancy of each mooring will be measured by comparing the results obtained from the assimilation with and without that particular mooring. While some degree of redundancy is necessary in operational observational systems (to provide a better estimate of the parameters to be observed and to minimize the detrimental effects of the failure or vandalism of one or more instruments), we will query if any mooring provides the same information as the rest of the array.

The outline of the paper is as follows. In section 2, we will describe the data used in this study. The ocean models (linear and nonlinear) are described in section 3. The optimal analysis using the linear ocean model is discussed in section 4. Section 5 introduces the simplification of the array. The last section contains a final discussion and conclusions.

## 2. Data

Observations of SSH come from TOPEX/Poseidon and Jason (TPJ) data, with a 17-cm bias correction applied to the Jason observations (G. Mitchum 2005, personal communication). The processing of the data is described in appendix B of Lagerloef et al. (1999). As a summary, the data are preprocessed along track to remove errors associated with tides, long wavelength, and high frequency signal due to orbit and environmental errors (such as ionospheric or water vapor corrections to travel time). Along-track data are filtered with a low-pass filter and binned every 0.25° and every 1/36 of a year. Operationally, TPJ anomalies are defined as sea level deviations from a 9-yr (1993–2001) mean sea level, which removes the geoid error. Next, the along-track data is gridded spatially using an optimal interpolation technique that takes into account grid location and propagation speed in the sea level signal. The 1993–2003 monthly anomalies are constructed with respect to the 1993–2003 seasonal cycle.

The blended SST analysis of Reynolds and Smith (1994) is used to estimate the spatial and temporal variability of the observed SST signal. The analysis is an optimal interpolation of all in situ temperature reports from ships, buoys, and satellite retrievals. The analysis produces a weekly mean gridded product on a 1° × 1° grid. A weekly climatology is constructed from the period 1993–2003, which is used to calculate weekly anomalies of SST.

Figure 1 displays the long-term mean (top), the standard deviation of the total signal (middle), and the 10-day anomalies (bottom) of both TPJ (left) and Reynolds SST (right) for the period October 1992–December 2003. Superposed on the fields is the spatial distribution of the proposed array of observing moorings (black dots). The long-term mean of sea level (Fig. 1a) is nonzero because of the disparity between the length of the time series used to estimate the seasonal climatology (1993–2003) and the period used to operationally estimate the geoid (1993–2001).

Mean sea levels show the deepening of the thermocline due to the ITF and the meridional convergence in the subtropical gyre of the Southern Hemisphere; the thermocline dome near 10°S, 55°E; and the smaller recirculation gyres off the western coasts of Africa and India. The reader is referred to Schott and McCreary (2001) for a review. The standard deviation of the total signal of sea level (Fig. 1b) shows the variability associated with the annual Rossby waves in the southern Arabian Sea and the Southern Hemisphere around 12°S (e.g., Perigaud and Delecluse 1993; Masumoto and Yamagata 1996). The spatial distribution of the Southern Hemisphere maximum at 10°S is in agreement with the amplitude of the annual harmonic of T/P estimated by (Wang et al. 2001). The northern signal corresponds to westward-propagating Rossby waves radiated from the western coast of the Indian subcontinent, which cross the Arabian Sea in about 3–4 months (Subrahmanyam et al. 2001). The maximum off Sumatra in both the standard deviations of sea levels echoes the anomalous climate events that took place in the IO during 1994 and 1997 (Murtugudde et al. 1999, 2000; Grodsky et al. 2001).

Mean SSTs denote the characteristic of the Indian Ocean in terms of an eastern warm pool and the western upwelling (Murtugudde and Busalacchi 1999). The western half of the Indo–Pacific warm pool covers much of the tropical IO (Fig. 1d) with average temperatures above 27°C. The standard deviation of total SST (Fig. 1e) highlights the seasonal modulation of the off-equatorial SSTs, while the standard deviation of the monthly anomalies reflects the cited 1994 and 1997 events. The difference in the color scales in Figs. 2e and 2f illustrates the minor contribution of the interannual variability toward the total variability of SST in the region (see Annamalai et al. 2003) even though high mean SSTs imply that these relatively small SST anomalies may still be climatically important.

Comparing the proposed array (black dots in Fig. 1) with the variability maps of SSH and SST, it can be seen that the array will be able to sample the regions for both seasonal and interannual variability of SSH (Figs. 1b and 1c). At first glance, the proposed array appears to cover, totally or partially, the main features of the IO variability. Indeed, the proposed mooring locations were chosen to capture some of the known features in the IO such as the east–west variability associated with the Indian Ocean Dipole/zonal mode (Murtugudde and Busalacchi 1999; Saji et al. 1999; Webster et al. 1999), the coastal upwelling off Java and Sumatra (Murtugudde et al. 2000; Susanto et al. 2001), the southwestern tropical IO associated with the thermocline doming (Reverdin et al. 1986), and the intraseasonal variability in the Bay of Bengal associated with the Asian monsoon (Webster et al. 1998; Sengupta and Ravichandran 2001). However, some of these features are only partially sampled, as the coastal upwelling off Java and Sumatra, or poorly sampled, as the westernmost variability off the African coast. Since the array is still in its design phase, simulated experiments may be used to assess the impact of these deficiencies and may help improve the final implementation in terms of its ability to capture the time and space scales of IO variability and its utility for climate prediction.

## 3. Models

The assessment of the proposed IO array will be carried out using two different versions of reduced-order Kalman filters incorporated into two different ocean models. The first part of the study, in which we will identify the optimal placement of the moorings, requires the analysis of millions of array combinations. However, the problem may be simplified by using a full version of the KF, as all the required information is contained in the error covariance matrix. A detailed explanation can be found in Hackert et al. (1998). The optimal application KF is restricted to linear models with a relatively small number of variables. The model chosen for this study is the linear ocean model of Cane and Patton (1984), which has already been used in similar studies for the Pacific and Atlantic Oceans (Miller et al. 1995; Hackert et al. 1998). Thus, the first questions to be addressed in this study are: Does the proposed array provide a reasonable sampling to recover the large-scale dynamical features resolved by a wind-driven, linear ocean model of the region when a quasi-optimal data assimilation method is applied? Does the position of the proposed moorings agree with the optimal locations selected by a linear ocean model?

The second part of this work is intended to assess the degree of redundancy of the proposed array. Any study of redundancy must take into account the inherent correlation between the different parameters being observed. For example, while the equatorial SST variability in the IO is driven by surface heat fluxes rather than ocean dynamics, there are regions, such as the southwestern part of the basin, where the correlation between monthly anomalies of SSH and SST is as large as 0.7. Moreover, occasional eastern upwelling develops on the equatorial IO, triggering ocean–atmosphere feedbacks coupling SST with ocean dynamics (Saji et al. 1999; Webster et al. 1999; Murtugudde et al. 2000). As the linear model to be used in the optimal localization of the moorings does not account for temperature variability, this part of the study will be based on the primitive equation, reduced-gravity, nonlinear model of Gent and Cane (1989). This part of the work will use a suboptimal data assimilation scheme based on projection of the analysis equations of the KF onto the multivariate empirical orthogonal functions (MEOFs) of the outputs of the model (see, e.g., Ballabrera-Poy et al. 2001). Disregard of the error subspace orthogonal to the MEOFs prevents the use of the estimated error covariance in assessing the impact of each mooring, and the importance of each mooring is assessed by calculating how its removal affects the reconstructed field (section 5a). Thus, proposing the removal of any mooring should be carefully assessed by ensuring the robustness of the methodology. A brief description of the linear and nonlinear ocean models is provided below.

### a. Linear model

The full version of the KF filter is incorporated into a coarse version of the linear ocean model of Cane and Patton (1984). The model state variables include the normalized height anomaly, zonal current, and Kelvin wave amplitude for each of the first two baroclinic modes. The model is set up for the tropical IO with a fairly idealized topography. The model domain spans from 30° to 20°S, 40° to 113°E with a resolution of ∼0.5° latitude and ∼1° longitude. Figure 2 shows the land mask for Africa, India, and Indonesia and the grid resolution for the model setup. The model is forced by the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996) wind anomalies with respect to the 1993–2003 mean. The winds are linearly interpolated to the 5-day time step of the model.

The model uses the linear shallow-water equations on the equatorial *β* plane subject to the long-wave approximation and finite differences in the horizontal directions to describe the evolution of each baroclinic mode. Linear damping is imposed by Rayleigh friction with decay times of 30 and 11 months for the first and second baroclinic modes, respectively. Kelvin wave speeds correspond to 2.53 and 1.56 m s^{−1}, *e*-folding length scales *L* ≈ (*c*/*β*)^{1/2} = 3.0° and 2.35°, and time scales (*T* ≈ (*cβ*)^{−1/2}) = 1.52 and 1.94 days, for the two baroclinic modes, respectively. Model wave speeds, length scales, and time scales are derived from density profile created using an average temperature and salinity profile from the *World Ocean Atlas* (*WOA*; Levitus and Boyer 1994) taken from a 10° box centered on the equator, 70°E. For each experiment, the model is spun up over the period January 1985–October 1992. Despite its inherent simplicity, the linear model shows significant correlations with SSH observations (larger than 0.7 as seen in Fig. 6a) in those regions where the seasonal and anomalous variances are large.

### b. Nonlinear model

The reduced-gravity, primitive equation, sigma-coordinate model with a variable-depth oceanic mixed layer is described in Gent and Cane (1989, hereafter GC) and Murtugudde et al. (1996). Vertical mixing is accounted for by using the hybrid mixing scheme of Chen et al. (1994). The model explicitly accounts for a complete upper-ocean hydrology (Murtugudde et al. 1998) with freshwater fluxes treated as a natural boundary condition (Huang 1993). An advective atmospheric mixed layer (Seager et al. 1995) has been coupled to the ocean model, which allows interactive surface heat flux and SST computations with no feedback to observations. Daily solar radiation and monthly cloudiness (Interannual Satellite Cloud Climatology Project, ISCCP), and weekly precipitation (Xie and Arkin 1998) are specified externally. This model has been used in a series of simulation studies of SST and circulation in all three tropical ocean basins (Murtugudde et al. 1996; Murtugudde and Busalacchi 1998; Murtugudde et al. 1998; Hackert et al. 2001). The ability of the model to reproduce the seasonal to interannual variability of the dynamics and thermodynamics of the IO has been reported before (Murtugudde and Busalacchi 1999; Murtugudde et al. 2000).

The configuration used for the present simulations covers the tropical Indian basin (30°N–30°S, 32°–140°E) with realistic coastlines and a homogeneous longitudinal and latitudinal grid spacing of ½°. The boundaries are treated as a sponge layer within 2° of the north and south boundaries smoothly relaxing to the *WOA* data. In addition, the mouth of the Persian Gulf, the Red Sea, and the Indonesian seas are also relaxed to the Levitus standards. The vertical structure consists of a variable-depth mixed layer and 24 sigma layers with a deep motionless boundary being specified as *T*_{bottom} = 6°C and *S*_{bottom} = 35 psu.

The model is spun up for 45 years, starting from rest with initial conditions derived from the *WOA* fields. During the spinup, the model is forced with a climatological version of NCEP winds (Kalnay et al. 1996), Xie and Arkin (1998) precipitation, and ISCCP clouds and shortwave flux, as well as the heat fluxes from the atmospheric mixed layer described above. The formula of Smith (1988) is used to convert the 10-m winds to wind stress. Interannual experiments for the period 1983–2003 are initialized from this climatological spinup and the wind speeds required for sensible and latent heat fluxes are computed using interannual wind stresses. Monthly climatological values are employed for cloudiness and radiation due to a lack of sufficiently long time series. An in-depth assessment of the ability of this model to represent the interannual variability of the IO may be found in Murtugudde and Busalacchi (1999) and Murtugudde et al. (2000).

## 4. Optimal mooring locations

### a. Methodology: Reduced-space Kalman filter

Application of a full KF in meteorology and oceanography is limited by its computational cost, associated with the size of the error covariance arrays and the number of model integrations required to compute their time evolution. Several studies have reduced the numerical cost of the KF by using a fine grid for the model and a coarse grid for the data assimilation (e.g., Fukumori and Malanotte-Rizzoli 1995). Miller et al. (1995) applied this approach to assimilate tide gauge and expendable bathythermograph data into the present linear model. This same technique was also shown to work in the tropical Atlantic (Hackert et al. 1998) for the goal of optimizing the locations of the PIRATA array. We will follow the nomenclature of Miller et al. (1995).

An optimal set of mooring locations is going to be identified by assimilating sea level data using a full KF applied on a coarse grid. Figure 2 compares the (finer) numerical grid of the ocean model with the (coarse) grid used in the data assimilation algorithm (i.e., the grid over which the diagonal of the error covariance matrix 𝗣 is defined). In the figure, the dots illustrate the grid spacing of the error covariance matrix while the cross hatching shows the full model grid. A series of sensitivity tests led to the conclusion that a 15 × 17 grid for the error covariance calculations in the KF is the optimal choice for our assimilation experiments since our main interest lies only in the large-scale features of the error field and the determination of a potential observing system. The resolution of the coarse grid is 5° in longitude and is stretched in latitude: 2° at the equator, 3° between 2° and 20°, and 4° between 20° and 30°.

In an initial reference experiment, sea level data at all 178 coarse-resolution grid points are assimilated using the reduced-order KF, which provides the time update for both the state of the system, **x**^{a}(*t*), and its expected error, 𝗣^{a}(*t*). During this process, the error covariance matrix rapidly reaches the stationary value 𝗣^{a}. Figure 3 shows the stationary value of the diagonal values of 𝗛𝗣^{a}𝗛^{T}(i.e., the expected sea level error variance at the coarse grid). The error structure illustrates the expected low error around the equator with a rapid increase of the error at higher latitudes, where information is being extrapolated. Exploiting the fact that for a linear system the Kalman filter provides the best linear unbiased estimate of the state of the system, the optimal selection of the moorings may be obtained using the statistical properties of the error covariance matrix. This method is based on the property that, if the geometry of the observing system is stationary in time, the time series of the differences **y**^{o}_{k} − 𝗛**x**^{a}_{k} are random variables of zero mean and covariance 𝗛𝗣^{a}𝗛^{T} + 𝗥. Therefore, the array 𝗣^{a} contains all of the information necessary to estimate the impact that changes in the observing system, 𝗛, will have on the error of the reconstructed state. Thus, the optimal array distribution is defined here as the subsampled set of observations that best reproduces the error structure of the reference solution.

### b. Results

A basic assumption is made that, for logistical reasons and to match the proposed mooring configuration, the IO mooring array will have a structure of five mooring lines. The location of these mooring lines is estimated by randomly subsampling the 178 × 178 error covariance matrix 𝗣^{a} at five arbitrary locations, leaving us with a 5 × 5 submatrix, 𝗛_{5}𝗣^{a}𝗛^{T}_{5}. Using this submatrix, the rest of the elements of the original error covariance matrix are estimated with a multivariate least square regression analysis. The analysis of the variance accounted for by each possible combination of five random moorings is used to determine the five locations that best reproduce the original error covariance matrix. The results consistently identify equatorial sites as the ones that carry most of the error information. This is not surprising considering that the linear model relies on equatorial dynamics to propagate large-scale features. The five equatorial locations that best reproduce the expected error are evenly distributed in longitude. Thus, the optimal array according to this linear data assimilative model consists of evenly spaced mooring lines at 55°, 65°, 75°, 85°, and 95°E. Those points are the equatorial black dots shown in Fig. 4.

Once the longitudinal location of the mooring lines has been selected, a second set of analyses is used to determine, one at a time, which of the moorings along the determined lines are optimal. Specifically, one point off the equator is chosen along with these five equatorial points from a pool of the available 173 mooring candidates. The ensemble of six points is then used to estimate the error field of the remaining 172 (i.e., 178 − 5 − 1) points. Again, the location with the highest average explained variance is chosen as an optimal location. In this way, the 35 optimal points are chosen in an objective fashion by using the forecast error field of the Kalman filter results. The 35 optimal points are shown in Fig. 4 (dots) along with the proposed sites (squares).

Comparison between the proposed and the optimal array shows that there are more similarities between both arrays than differences. The main difference is associated with the longitudinal regularity proposed by the optimization algorithm, the reduced number of moorings in the 55°E mooring line, and the lower latitudinal resolution at the equator. The optimal array does not identify the need for an additional sampling at 110°E, which is directly related to the fact that the error covariance matrix only accounts for sea level since the model does not include the variability of the temperature field and the nonlinear effects of upwelling off Java. The Ekman pumping due to the wind stress curl and the thermocline doming in the southwest (Reverdin et al. 1986; Murtugudde and Busalacchi 1999) are also not captured by the linear model. The intraseasonal variability in the Bay of Bengal along 90°E is not evident in the low frequency variability of the SSTs or sea levels and the low frequency forcing of the linear model is unable to simulate the intraseasonal variability. Even in an OGCM, special care will be required to optimize the mooring array in this region.

Despite this fact, the agreement between both arrays is striking in the relative amount of moorings that both arrays propose for the different longitudinal lines. The geographical distribution around 65°E is very similar. Using the same number of moorings, the coarser meridional resolution around the equator translates into a larger meridional extension. Note that the optimal array proposes additional moorings in the Arabian Sea. The proposed line at 80°E is mirrored with two lines with very similar latitudinal extension at 75° and 85°E. Finally, the optimal line at 95°E combines the longitudinal sampling of the proposed lines at 90° and 95°E. Note that Rossby waves dispersing off of the southwest corner of India into the Arabian Sea and the equatorial Kelvin waves traveling up into the Bay of Bengal as coastal Kelvin waves are likely driving the meridional extensions of the optimal array lines along 65° and 95°E, respectively (see, e.g., Murtugudde et al. 1999).

The theoretical and practical advantages of both array configurations are illustrated in Figs. 5 and 6. The comparison of the amount of variance of 𝗣^{a} explained by both array configurations is shown in Fig. 5. The differences between the proposed minus the optimal explained variance indicate that the optimal array does better (blue) in the Arabian Sea, while the proposed array has an overall higher explained variance (red) in the Bay of Bengal and along 10°S stretching across the basin. The average mean for the differences is negative (−0.16%), indicating that the optimal configuration has overall slightly higher explained variance, as expected from the maximization procedure used to select the moorings. The knowledge of specific climatic features such as the thermocline–mixed layer interactions in the dome region around 10°S, 55°E and the intraseasonal variability in the Bay of Bengal should obviously supersede the conclusions of the linear model.

To show the impact the different array configurations, experiments that assimilate observed SSH at the optimal and proposed mooring sites were completed. Figure 6 displays the correlation fields (top) and rms of the difference (bottom) between model results and observed sea level. Three experiments are shown. Figure 6a corresponds to the correlation and rms error when no data is assimilated into the linear ocean model. Figure 6b corresponds to the case where data is assimilated following the sampling derived from the optimal procedure. Finally, Fig. 6c corresponds to the correlation and rms error obtained when the data sampling comes from the proposed array. The results shown in Fig. 6b indicate the good coverage of the simulated mooring sites and the expected improvement over the case with no assimilation. The comparison between Figs. 6b and 6c shows the positive impact of the extra sampling in the central Arabian Sea. Conversely, note that the sampling of the Bay of Bengal in the optimal configuration is not able to produce a low error estimate in the central part of the bay. In this region, the sampling line at 90°E is shown to produce overall better results in the Bay of Bengal. Also note that there is no appreciable improvement in the proposed array correlation and rms difference (Fig. 6c) near the equator due to increased sampling at 2°N and 2°S with respect to the assimilation at optimal locations (Fig. 6b). This is expected since there are no sharp meridional gradients in the near-equatorial IO, as seen in Fig. 1. This redundancy is also highlighted in the second approach discussed in the next section.

The overall similarities between both the proposed and the optimal array must be seen as a first justification of the sampling proposed by the IO Panel. However, the inherent limitations of the linear model to reproduce off-equatorial processes may impede use of the optimal method to propose moorings south of 15°S. Note the disparity between the smooth increase of the analysis error south of 15°S shown in Fig. 3 with the rapid increase of the error shown after the actual assimilation (Figs. 6b and 6c). Thus, these results should be considered as a rationale to discuss further the higher latitudinal resolution present in the proposed array.

## 5. Array simplification

The similarities between the optimal distribution of moorings and the subjectively proposed array shown in the previous section provide a strong justification for the array proposed by the IO Panel. Therefore, the second part of this work will focus on the issue of simplification of the proposed array rather than the optimal array described in the previous section. To test if the array might be simplified, we investigate how the basinwide reconstruction of the field is affected after the removal of each individual station. After removing each station, the corresponding reconstructed fields of sea level and SST fields are compared against the observed data. Comparison against model-generated fields could lead to an overly optimistic assessment of the ability of the moorings to reconstruct the basinwide model fields and lead to a misleading oversimplification of the mooring array. Basinwide reconstruction of the fields requires extrapolation to regions outside the span of the array. The error map shown in Fig. 3 shows that the largest increase in the expected error is located in those regions where information is being extrapolated (i.e., south of 15°S). To reduce the impact of extrapolation errors in our assessment, we limit the size of the study region to the band 15°S–15°N (i.e., we do not consider the two stations outside this band, and the proposed array to be considered here has 33 stations).

### a. Methodology: Fitting observations onto the MEOFs of the nonlinear model

The monthly averaged outputs of SSH, SST, sea surface salinity (SSS), mixed layer depth (MLD), and surface ocean currents from the GC model are used to construct a multivariate vector. Figure 7 shows the first EOF of the monthly anomalies of the multivariate vector. This mode accounts for 10% of the variance of the vector anomalies during these 31 years (1983–2003). The SSH and SST components of this MEOF show the spatial patterns of the IODZM. Moreover, the SSH and surface velocity patterns are similar to the spatial patterns described by Grodsky et al. (2001). The time modulation of this mode has significant maxima for the events of 1994 and 1997. The variability represented by this mode completely corresponds to interannual variability, as the first significant maximum has a quasi-biennual periodicity. Interestingly, the time series of the first model also shows a significant maximum during 1986. The origin of such a signal is not clear, as it can respond to any of the variables of the multivariate function. Investigation of such a feature goes beyond the scope of this work.

Thus, fitting the TPJ SSH and Reynolds SST observations, subsampled at the mooring locations, to the MEOFs of the GC model allows reconstruction of basinwide fields of SST, SSH, SSS, MLD, and surface currents. The algorithm used here is described in Ballabrera-Poy et al. (2001). Additional technical details may be found in Smith et al. (1996) and Oke and Schiller (2007). The observational errors of SSH and SST are supposed to be uncorrelated and constant through the whole system.

*J*, which accounts for the combined (SSH and SST) rms between the reconstructed field and the observations:

*a*” and “

*s*” indicate the analysis and satellite-estimated fields (both defined on the same model grid). Each term is normalized by the number of corresponding observations to avoid biasing the cost function toward the field with the largest amount of observations. Since we are interested in the ability of the array to reconstruct the variability of the system in the tropical region, the sum in (1) is restricted to the latitudinal band between 15°S and 15°N. In this case,

*n*

_{SSH}= 6040 and

*n*

_{SST}= 6466. The difference in the number of ocean points is due to the different ocean–land masks of both satellite estimates. Note that Eq. (1) provides a rather terse description (a single number) of the two-dimensional error field of SST and SSH. While such simplification obviously lacks an appropriate representation in the error, it is required in order to apply any automatic optimization technique. Although the MEOFs used in the assimilation algorithm also lead to the reconstruction of basinwide estimates of SSS, and surface currents, only SST and SSH are being used in Eq. (1) as they are the only physical parameters being currently monitored.

Figure 8 shows the time evolution of *J*(*t*) (and the sea level and temperature contributions, *J*_{SSH} and *J*_{SST}) when all 33 proposed moorings are assimilated. The horizontal line at 0.5 approximates the value obtained if the spatial variance of the analysis error equals the observational error variance. Since the value of the cost function (1) is obtained using observations that have not been assimilated, the sensitivity of (1) to some assimilation parameters may be used to adjust them. For example, experiments using 20 MEOFs deliver values of the cost function as large as 1.8. Increasing the number of modes reduces the value of the cost. However, after some initial drastic reduction, increasing the number of modes hardly reduces the cost. Figure 8 and the experiments described below are obtained using 35 MEOFs. On the other hand, Fig. 8 indicates that the reconstructed sea level has a larger relative error than temperature. This indicates that the system has a tendency to overfit temperature. The disparity is even more prominent if the observational error of sea level and SST are set to 10% of the observational range. Increasing the SST observational error reduces both the value of *J*_{SST} (expected as the observational error is in the denominator) and *J*_{SSH}. While observational errors larger than 0.8°C still reduce the value of *J*_{SSH}, their impact on *J*_{SST} is much more dominant, further increasing the disparity between J_{SSH} and J_{SST}. Figure 8 is obtained using *ε*_{SSH} = 7 cm and *ε*_{SST} = 0.8°C.

The most remarkable feature of Fig. 8 is the large analysis error toward the end of 1997, during the event described, for example, by Murtugudde et al. (2000) among others. The eigenvalue analysis of the error fields shows that the error committed during this event accounts for more than 22% of the error for both SSH and SST. Figure 9 compares the Hovmöller diagrams of equatorial observations (left) and analysis (right). The plots show that the analysis fields reproduce observed attributes, specially the propagation feature during 1997–98 and the strong anomalies associated. The nature of the MEOFs precludes the reconstruction of the high-frequency patterns in the western part of the basin. The ocean current obtained from the MEOFs (not shown) exhibits the westward currents required to maintain the equatorial gradient of SST during these 1994 and 1997 events.

To identify the most redundant mooring, a series of 33 data assimilation experiments is performed. For each experiment, one single mooring is removed systematically. Equation (1) is calculated and compared against the cost function from the full array. It is expected that the assimilation of data from each abridged array will generate estimates of the state of the IO with a larger error than when the full array is assimilated, and this will be seen as an increase in the value of the cost function (1). The larger the error increase, the larger the value *J* will be. Thus, the most redundant mooring will be the one whose removal provides the lowest change in the value of *J*.

### b. Results

Figure 10 shows the temporal average of *J*(*t*) as a function of the mooring being removed. In this plot, the zero abscissa indicates the experiment where no moorings have been removed, abscissa one indicates the experiment for which the mooring labeled one (12°S, 55°E, as seen in Fig. 4) has been removed from the full array, abscissa two indicates the experiment for which the mooring labeled two (8°S, 55°E) has been removed from the full array, and so on. The horizontal lines at 0.129 and 0.167 indicate the 90% confidence interval of the mean value when all moorings are assimilated. The three main characteristics of Fig. 10 are (i) the removal of a single mooring does not significantly change the time mean of the cost function, (ii) no absolute minimum may be singled out from the multiple minima, and (iii) the regularity of the minima indicates that the most redundant stations are the ones located near the equator.

The fact that the average value of the cost function does not significantly change as one mooring is removed indicates that, with respect to this measure, no mooring contains exclusive information of relevance to the basinwide, monthly averaged, variability of SSH and SST. That is, the proposed array contains the degree of redundancy appropriate to guarantee the reconstruction of these fields after the accidental removal of one of their components. Note that this does not imply that they are all redundant. It just corroborates that the spacing between moorings is not significantly larger then the decorrelation scales of the processes being reconstructed. Figure 10 also reveals that, in general, outer moorings carry the less redundant information, while inner moorings are the most redundant. However, the absolute minimum (when one station is being removed) is obtained when station 27 (8°N, 90°E) is withdrawn. This means that the information from this outer northern mooring does not provide anything new, once the rest of the moorings are taken into account, to the variability that influences the value of *J*.

The equivalence between all local minima means that not a single mooring location can be singled out as completely redundant and that, after this first experiment, there is no objective basis to discard one of the moorings for providing utterly redundant information. The smoothness around each local minima in Fig. 10 indicates that, in every case, the most redundant mooring might be considered either the one at the equator (e.g., the minimum at B, i.e., 0°, 67°E) or one of the nearby moorings (1.5°S or 1.5°N, 67°E). This fact may be explained by the increased meridional resolution on each side of the equator.

To further investigate if any mooring might be removed from the proposed array, five families of additional experiments are carried out by further reducing the size of the array. Each one of these five families of experiments (named A, B, C, D, and E, corresponding to the five local minima shown in Fig. 10) begins with a 33 − 1 mooring array. For example, the 32 initial moorings of Family A are obtained by removing mooring number 5 (the westernmost local minimum) from the complete set of moorings. Starting with this reduced array, 32 data assimilation experiments are performed by removing one additional mooring. The mooring whose removal gives the lowest cost function is identified and removed, providing a 31-mooring array. The process is repeated until discernible differences in the cost function are identified. The same procedure is repeated for the other four families. Table 1 shows the order in which the arrays are being removed during each experiment. Shown is the number that identifies each mooring and its spatial location. After five experiments, Families B, C, and D converge. Five moorings, highlighted in bold, share two properties: (i) these moorings are the first to be automatically removed from each experiment (moorings 5 and 32 were removed “by hand” because they are local minima in Fig. 8) and (ii) the order in which these five moorings are removed changes in each experiment. The five moorings highlighted are 12, 20, 21, 26, and 27. The differences in the order these five moorings are removed indicates that it is not possible to single out any one of them as being the most redundant.

The above discussion indicates that there is no evidence to pick one mooring from the set of the redundant moorings. But a question remains: Are these moorings really providing redundant information? To answer this question, the standard deviation of the difference between the 33-mooring assimilation and the 32-mooring assimilation (where station 27 has been removed) has been calculated. The largest values of the standard deviation for SSH and SST are 0.5 cm and 0.06°C. These differences are one order of magnitude smaller than the respective observational errors. In other words, mooring 27 provides only redundant information with respect to the basinwide reconstruction of monthly anomalies of SSH and SST. However, the discussion in the previous paragraph tells us there is no way to pick one over the others. On the other hand, if the five most redundant moorings are removed, the standard deviation of the difference against the 33-mooring assimilation becomes larger than the observational error. Thus, removal of the five most redundant moorings translates into a significant loss of information. These results show that, although some moorings have been shown to only provide redundant information, there is no basis for removing one rather than the others and no mooring can be identified as being expendable.

## 6. Discussion and summary

Objective methodologies have been applied to the selection of the optimal locations for a mooring array in the tropical IO. A full Kalman filter is incorporated into a linear ocean model to assimilate altimeter sea levels and satellite-derived SSTs. The objective locations are chosen using maximum average explained variance of the reconstructed error field for the whole basin as a metric to successively optimize a mooring array configuration. In many respects the array configuration chosen by the objective technique closely matches the subjective array. However, the mooring locations proposed by the CLIVAR IO Panel do a better job of recreating the observed signal in the Bay of Bengal and along 10°S, whereas the optimally derived sites better reproduce the error in the Arabian Sea. While the data assimilation technique with the full Kalman filter is clearly optimal, the model into which observations are assimilated is decidedly simplistic and may preclude the method from pointing out the need for additional mooring stations near the coast off Sumatra and Africa as well as the need for increasing the sampling south of 15°S. A generalization of these experiments to a more sophisticated ocean model is of obvious difficulty, both because of its computational cost and the increased difficulty in parameterizing the model error. Moreover, the assimilation of vertical profiles of temperature requires an adequate paradigm for balanced updates of vertical profiles of the salinity and velocity fields (see Vossepoel et al. 2002). We thus deem the choice of this simple model with a realistic assimilation technique to be a good compromise as a first step toward offering guidance for the implementation of the IO observational array.

The ocean dynamics in the IO is dominated by multiple recirculation zones and the semiannual Wyrtki jet on the equator with no strong evidence of sustained mixed layer–thermocline interactions or strong equatorial upwelling (see Schott and McCreary 2001; Annamalai and Murtugudde 2004). Thus, capturing the Indian Ocean SST and heat content variability may be relatively simpler compared to the tropical Atlantic and Pacific Oceans, even though the SST variance at all time scales tends to be small and mostly in the range of observational errors, making selecting an array more challenging.

Simplification of the proposed array has also been investigated using the statistical properties of a primitive equation model summarized by its MEOFs and fitting the observed data, sampled at the array locations, onto these modes. The results indicate that, although equatorial moorings display the largest amount of redundant data and the most redundant moorings have been shown to provide no new information, no mooring can be identified as expendable for reconstructing the basinwide SSH and SST. The smoothness of local minima A, B, and C, and the maintenance of the basic sensitivity of the array as equatorial moorings are being removed can be explained by the enhanced latitudinal resolution near the equator. Note that the objective technique never identifies 2°N and 2°S (the closest points to the equator in the coarse grid) as optimal, even though linear model dynamics might suggest otherwise. We have performed several additional experiments by changing the number of MEOFs to be reconstructed and by using a Markov model to account for the temporal persistence of the MEOF coefficients as in Kaplan et al. (2000). In all cases, these experiments point out that the most redundant stations are the near-equatorial ones. Our results do not justify the exclusion of any mooring. Instead, they indicate that more work needs to be done to pinpoint how prioritization in implementing the elements of the array can be undertaken. For example, in the context of SSH and SST fields, the same number of stations could potentially be used to sample a larger latitudinal region rather than resolving latitudinal gradients at the equator.

An OSSE to resolve intraseasonal variability may lead to other features such as the formation of the monsoon onset vortices and the warm pool off southwestern India prior to the onset of monsoons (Shenoi et al. 2002). On the other hand, fine-tuning the implementation of an array based on simple model OSSEs has to take into consideration all of these caveats. The multimodel OSSEs as planned under the CLIVAR IO Panel will, in fact, enhance the overall value of the OSSEs.

We expect that our study, along with other parallel OSSEs with other models and cost functions, will yield a body of information that will lead to optimization of the IO array and allow phased implementation to achieve the overall goals of the observation system as described in the relevant CLIVAR science plans.

## Acknowledgments

We wish to thank NCEP for providing their wind data. We would also like to thank Gary Mitchum for providing the gridded TOPEX/Poseidon and Jason sea level data. We appreciate the comments of the reviewers, which helped to improve the paper significantly. This research is supported by a NASA Jason grant.

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Model grid (hatching) for the linear experiments and the error grid (black dots).

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Model grid (hatching) for the linear experiments and the error grid (black dots).

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Model grid (hatching) for the linear experiments and the error grid (black dots).

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Forecast error covariance (𝗛𝗣𝗛^{T}) for the experiment that assimilates sea level data every 2° lat and 5° lon.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Forecast error covariance (𝗛𝗣𝗛^{T}) for the experiment that assimilates sea level data every 2° lat and 5° lon.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Forecast error covariance (𝗛𝗣𝗛^{T}) for the experiment that assimilates sea level data every 2° lat and 5° lon.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Locations of OPTIMAL (solid circles) and PROPOSED ARRAY (open squares) mooring sites: numbers correspond to the names of the proposed moorings used in section 5.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Locations of OPTIMAL (solid circles) and PROPOSED ARRAY (open squares) mooring sites: numbers correspond to the names of the proposed moorings used in section 5.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Locations of OPTIMAL (solid circles) and PROPOSED ARRAY (open squares) mooring sites: numbers correspond to the names of the proposed moorings used in section 5.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Difference between optimal interpolation of optimal minus proposed mooring sites for percentage explained variance: black (white) dots indicate optimal (proposed) mooring sites.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Difference between optimal interpolation of optimal minus proposed mooring sites for percentage explained variance: black (white) dots indicate optimal (proposed) mooring sites.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Difference between optimal interpolation of optimal minus proposed mooring sites for percentage explained variance: black (white) dots indicate optimal (proposed) mooring sites.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

(top) Correlation and (bottom) rms differences between data assimilation or model results and observed sea level for (left) NOASSIM: model alone; Fig. 6 (continued) (middle) assimilation at optimal locations, and (right) assimilation at proposed mooring site locations. Black dots indicate assimilation locations for each figure.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

(top) Correlation and (bottom) rms differences between data assimilation or model results and observed sea level for (left) NOASSIM: model alone; Fig. 6 (continued) (middle) assimilation at optimal locations, and (right) assimilation at proposed mooring site locations. Black dots indicate assimilation locations for each figure.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

(top) Correlation and (bottom) rms differences between data assimilation or model results and observed sea level for (left) NOASSIM: model alone; Fig. 6 (continued) (middle) assimilation at optimal locations, and (right) assimilation at proposed mooring site locations. Black dots indicate assimilation locations for each figure.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

(a)–(d) First multivariate EOF of the GC ocean model, (e) its normalized time series, and (f) the power spectrum. This mode explains 10% of the variance of the multivariate vector (SSH, SST, SSS, MLD, *U*, and *V*): units of the spatial loads are the standard deviation corresponding to each field.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

(a)–(d) First multivariate EOF of the GC ocean model, (e) its normalized time series, and (f) the power spectrum. This mode explains 10% of the variance of the multivariate vector (SSH, SST, SSS, MLD, *U*, and *V*): units of the spatial loads are the standard deviation corresponding to each field.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

(a)–(d) First multivariate EOF of the GC ocean model, (e) its normalized time series, and (f) the power spectrum. This mode explains 10% of the variance of the multivariate vector (SSH, SST, SSS, MLD, *U*, and *V*): units of the spatial loads are the standard deviation corresponding to each field.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Time evolution of the cost function (1) when all 33 stations are being assimilated: shown are the total cost (black) and the SSH (red) and SST (blue) components. The cost function is unitless.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Time evolution of the cost function (1) when all 33 stations are being assimilated: shown are the total cost (black) and the SSH (red) and SST (blue) components. The cost function is unitless.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Time evolution of the cost function (1) when all 33 stations are being assimilated: shown are the total cost (black) and the SSH (red) and SST (blue) components. The cost function is unitless.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Longitude–time evolution of (left) observations and (right) analysis of (top) SSH in m and (bottom) SST in °C.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Longitude–time evolution of (left) observations and (right) analysis of (top) SSH in m and (bottom) SST in °C.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Longitude–time evolution of (left) observations and (right) analysis of (top) SSH in m and (bottom) SST in °C.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Sensitivity of the time average of the cost function when a single mooring is being removed. The horizontal axis indicates the mooring being removed. The spatial location of the moorings is given in Fig. 4 (e.g., mooring 5 corresponds to 0°, 55°E). The origin 0 corresponds to the experiment where all stations are being assimilated.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Sensitivity of the time average of the cost function when a single mooring is being removed. The horizontal axis indicates the mooring being removed. The spatial location of the moorings is given in Fig. 4 (e.g., mooring 5 corresponds to 0°, 55°E). The origin 0 corresponds to the experiment where all stations are being assimilated.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Sensitivity of the time average of the cost function when a single mooring is being removed. The horizontal axis indicates the mooring being removed. The spatial location of the moorings is given in Fig. 4 (e.g., mooring 5 corresponds to 0°, 55°E). The origin 0 corresponds to the experiment where all stations are being assimilated.

Citation: Journal of Climate 20, 13; 10.1175/JCLI4149.1

Order that each individual station is removed from each family of assimilation experiments. Each family letter corresponds to a local minimum shown in Fig. 10. Highlighted in bold font is the first station removed from each experiment. Note that the order of the removal of these five stations changes from experiment to experiment