1 INTRODUCTION

Ocean waves are usually predicted by running numerical wave models. The skill of a numerical wave forecast (NWF) depends on the accuracies of the computer model and the boundary and initial conditions. The accuracy of the model depends on our underst and ing of the physics of wind-wave processes. Boundary conditions focus on surface forcing (sea surface winds) and lateral forcing (which mainly refers to incoming ocean swells across open boundaries). Besides these two aspects, NWF is an initial-value problem. A more accurate initial wave field will produce a more accurate forecast. Initial conditions are provided by wave observations. Traditional wave forecasts use a numerical wave model to produce a forecast, and then adjust the parameters according to observed data. In contrast, we use the numerical wave model to forecast waves after finding the best parameters. Developments in satellite remote-sensing technologies and a significant amount of daily altimeter observations mean that we may be able to improve the quality of the initial fields using data assimilation (i.e., optimally combining information from observations and information from dynamic models). We have daily along-track observations from satellite altimeters, and they are sufficiently dense for data assimilation.

Data assimilation for wave models began in the late 1980s, but was not widely applied in operational practice until the 1990s, in a few Western countries. Data assimilation methods can be divided into two general classes: variational and sequential. To summarize their differences, the sequential (or ‘single-time-level’) method does not consider the model dynamics, whereas the variational method explicitly uses the model dynamics in the data assimilation process (Greenslade, 2001). The variational method that has been used in meteorological forecast models for many years ( Le Dimet and Talagrand , 1986; Talagrand and Courtier, 1987) is based on optimal control theory. It attempts to minimize the given cost function, which measures the misfit between the model and observations, by optimizing the control parameters. In numerical weather prediction, variational data assimilation helps improve the quality of the forecast by improving the initial conditions.

Various sequential methods have been proposed since Thomas (1988) and Esteva (1988). Among them, optimal interpolation (OI) was proposed first and is the most widely used in operational practice. Lionello et al. (1992) assimilated significant wave height (SWH) data from the US first Earth-orbiting satellite SEASAT and the Navy satellite GEOSAT altimeters into the WAM (Wave Modeling) model using OI, and reconstructed the analyzed wave spectrum using a method similar to Thomas (1988). Subsequently, Lionello et al. (1995) successfully applied this scheme to global wave analysis and prediction. Hasselmann et al. (1997) and Voorrips et al. (1997) divided the full spectrum into several systems, each represented by one of three parameters. They used this assimilation scheme with observed partition parameters. Bender and Glowacki (1996)applied this scheme to the Australian version of WAM, incorporating GEOSAT SWH observations into the model grids using statistical interpolation. They demonstrated a 50% reduction in the root mean square error (RMSE), which means that this method is feasible for operational applications. Greenslade (2001)improved the above work using more realistic wind fields and a more extensive set of independent wave rider buoys. They discussed the application of the well-documented spectral adjustment method. Zhang et al. (2003) applied this scheme into typhoon waves in the South China Sea. Wang and Yu (2009) validated the impact of altimeter SWH data assimilation on wave models in the northwest Pacific, and achieved a remarkable improvement in the 0-24 h wave forecasts. The results of all of these studies suggest that the OI method is an efficient and attractive data assimilation technique for operational wave forecasting.

Generally, data assimilation in the OI scheme assumes that the background error covariance (BEC) is temporally stationary and spatially uniform. In fact, the spatial structure of the BEC should be determined according to complex physical processes, and should be considered inhomogeneous in space and unsteady in time. An inaccurate estimate of the BEC is the major hurdle of the OI based assimilation technique. Many researchers have pointed out this limitation. Greenslade and Young (2004) calculated the correlation between modeled SWHs and biascorrected ERS-2 data. They suggested that the length scale of the background errors had a significant spatial context (with the largest scales at low latitudes and shortest scales at high latitudes) and were relatively steady in a temporal context. However, the magnitudes of the errors had considerable seasonal signals. Similar results were presented by Greenslade and Young (2005) and Greenslade et al. (2005). Guo et al. (2012) simulated the BEC of SWH fields using the Monte Carlo method, and concluded that it varies consistently with the mean wave direction (MWD). Based on the correlation between the BEC and the MWD, a new BEC model of SWH was developed. Their case study of regional data assimilation showed that the developed BEC model produced better wave forecasts and reasonable approximations of anisotropy and inhomogeneous errors.

Despite these weak points, the OI has been widely used in operational practices because it is relatively easy to implement and has low computational costs.

The ensemble Kalman filter (EnKF) and smoother (EnKS) are other types of sequential methods. They have been extensively investigated in the context of nonlinear ocean dynamics (Evensen, 1994; Burgers et al., 1998; Evensen and van Leeuwen, 2000, etc.). The EnKF and EnKS schemes use Monte Carlo forecasting or ensemble integration to compute the time evolution of error statistics. The Monte Carlo method avoids computational problems associated with the traditional extended Kalman filter (EKF), by neglecting contributions from high-order statistical moments when solving the error covariance equation. Ensemble methods require the evaluation of dynamics for a large number of statistical ensembles, so coupling this surrogate (instead of a full model) with the EnKF improves the computational efficiency. Based on this, Zamani et al. (2010)presented a dynamic artificial neural network (ANN-type) windwave model and an EnKF, which had a modest execution time.

Localization is a necessary feature of data assimilation. Spurious long-range correlations may exist because of the limited sample of ensemble members, so we should limit the horizontal extent of the updates (Counillon and Bertino, 2009). Localization allows for a local analysis at a grid point using only nearby observations. To implement localization, Evensen (2003) suggested multiplying innovations (differences between forecasts and observations) by a step function.

Ensemble optimal interpolation (EnOI) was proposed as a suboptimal method, when compared with the EnKF (Evensen, 2003). EnOI is an attractive approach if we wish to save computational time. After creating the stationary ensemble, we only need one single model integration in addition to the analysis step, and the error statistics are invariant over time. The final update cost is reduced to 1/N (where N is number of ensemble members) compared with the EnKF. Thus, the EnOI method is numerically extremely efficient, but always provides a suboptimal solution when compared with the EnKF (Evensen, 2003). Another benefit of EnOI is that the ensemble error covariance (as an estimation of the BEC) reflects the length-scales and anisotropy of the model state fields. Additionally, EnOI can be used to readily assimilate different observation types in a single step (Oke et al., 2008). This contrasts with many other data assimilation schemes, which require special methods to assimilate satellite-derived sea-level anomalies (SLA) and in situ temperature (T) and salinity (S) (Cooper and Haines, 1996; Troccoli and Haines, 1999; Segschneider et al., 2000; Fox et al., 2002; Guinehut et al., 2004; Cummings, 2005; Chassignet et al., 2007; Martin et al., 2007). The EnOI has been applied in the current model to assimilate temperature and salinity fields (Fu et al., 2009; Xie et al., 2011). Yan and Zhu (2011) discussed the sampling strategy of the stationary ensemble in the EnOI using the current model.

Several modified wave data assimilation methods have recently been proposed. Zhang et al. (2006) combined statistical interpolation (SI) assimilation and artificial neural networks (ANN), mimicking the errors introduced by the wave model. Sannasiraj et al. (2006) updated wave parameters using the ensemble error covariance over the forecast horizon. They corrected the extended forecast using the observations and forecasts from “local models” (Babovic et al., 2005; Sannasiraj et al., 2005). To extend the extent of the improvement from the assimilation, Galanis et al. (2009) presented an approach that first estimates the wave model’s systematic errors using a combination of the Kolmogorov-Zurbenko and Kalman filters, and then assimilates these estimates during the forecast period using an improved classical statistical interpolation method. Based on this research, several studies investigated first using Kalman filtering algorithms (Walker, 2006; Emmanouil et al., 2010) or statistical Kalman filters (Emmanouil et al., 2012) and then applying the OI scheme.

In a region where swells are the principal component of the sea state, assimilation is more effective than in a wind-sea dominated area (Emmanouil et al., 2007). This is one reason for choosing the north Indian Ocean as our target computational domain.

For open oceans such as the Indian Ocean, satellite altimeter wave heights are obviously our first choice for data assimilation, because there are very few buoy observations.

The purpose of this study was to improve the accuracy of numerical wave forecasts using an EnOIbased assimilation of altimeter SWH data. We present the results of our preliminary investigation into the impacts of the EnOI-based assimilation of altimeter data on the initial field and forecasting results. The outline of the paper is as follows. Section 2 describes the assimilation method for the EnOI. We describe the wave assimilation system and experiments in Section 3. Section 4 contains the ensemble error correlations and the flow-dependency. Results from the assimilation experiment are assessed and compared with observations in Section 5. Section 6 provides a summary of the results.

2 ASSIMILATION TECHNIQUE 2.1 Ensemble optimal interpolation

As in Evensen (2003, 2004), ψ_j = (j =1, 2, …, N) are defined as members of the ensemble for a univariate model state and the assimilation variable, SWH. They are combined in a vector,

where N is the number of ensemble members.

The ensemble anomaly A' is then defined as

where A is the ensemble mean and 1_N ϵ ℜ^N×N is a matrix with each element equal to 1/N.

The ensemble error covariance matrix is

where T represents transposition.

Given an observation vector d ϵ ℜ^m (where m is the number of observations), the N vectors of perturbed observations can be defined as

where ε_j is the associated uncertainty or perturbations, and j = 1 … N. Then, the ensemble of perturbations (with a mean of zero) can be stored in an observation error matrix

from which we construct the ensemble representation of the observation error covariance matrix,

The analysis field in the EnOI scheme is computed by solving

where the model forecast and analysis are denoted as ψ_f and ψ_a, respectively. H is an operator that interpolates from the model grid to the observation locations. The parameter α∈(0, 1] weights the ensemble and observation errors, and is especially important when the ensemble has a climatological signal that contains a bias.

We should limit the horizontal extent of the updates because spurious long-range correlations can occur because of the limited sample of members (Counillon and Bertino, 2009). To implement localization, Evensen (2003) suggested multiplying the innovations ((d - Hψ_f) in Eq.7) by a step function. Similar to Counillon and Bertino (2009), we use a weight function for the innovations that introduces an exponential decay, and ensures a smooth transition at the edges of the localization area.

2.2 Wave spectrum adjustment

In the wave model, the wave spectrum is directly integrated in the full spectral space. That is, the wave model requires a full, two-dimensional wave spectrum for the assimilation. However, a satellite altimeter only provides observations for one integrated spectral parameter, the SWH. Because SWH is a model output variable rather than a model integration variable, the observed SWH data must be assimilated by adjusting the spectrum.

There may exist different wave spectra that can lead to a particular observed wave height. Hence, we must make some assumptions to derive the wave spectrum from SWH and assimilate it into the wave model. There have been attempts to derive analyzed wave spectra from the altimeter data. In this paper, we used the scaling approach mentioned earlier (Esteva, 1988) to assimilate the altimeter data. This approach ensures that the total energy under the modified spectrum will correspond to the observed SWH, while the shape of the energy distribution in terms of frequency and direction remains unchanged.

SWH is proportional to the integral of the wave spectral energy over all frequencies and directions. The analyzed spectrum at each grid-point F_α(σ, θ), where σ is the wave number and θ is the direction, is obtained from the predicted spectrum, F_f(σ, θ), with the ratio (ψ_a/ψ_f)². That is,

This technique was used in Esteva (1988) and Bauer et al. (1992). Both of these studies found a positive impact on the modeled wave fields.

3 SETTINGS FOR THE WAVE DATA ASSIMILATION SYSTEM AND EXPERIMENTS 3.1 Grids for the nested wave model

The wave model uses Version 3.14 of the fullspectral third-generation wind-wave model WAVEWATCH III (Tolman, 2009). The target computational domain for the data assimilation experiments was north of 15°S in the Indian Ocean and South China Sea (SCS), i.e., 15°S-30°N, 30°- 122°E, which covers the famous monsoon areas. Considering the incoming southern ocean swells that propagate thous and s of kilometers to our target domain through open boundaries, we used a two-way nesting strategy. That is, the high-resolution target domain was nested in a global (i.e., 78°S-78°N, 180°W-180°E) WAVEWATCH III implementation. Version 3.14 of WAVEWATCH III introduced a multigrid approach to wave modeling, allowing for a twoway nesting between grids.

The nested model used a 0.25°×0.25° spatial grid resolution for the target domain, and a 1.0°×1.0° resolution (in latitude and longitude) for the global domain. Bathymetry data from the ETOPO5 database (provided by U.S. National Geophysical Data Center) were used for both domains, and they used the same spectral grid setting.

The wave model was forced by surface wind velocities from the cross-calibrated multi-platform (CCMP) surface winds provided by the NASA PO. DAAC. This dataset was derived by cross-calibrating and assimilating ocean surface wind data from SSM/I, TMI, AMSR-E, SeaWinds on QuikSCAT, and SeaWinds on ADEOS-II. The CCMP winds were validated before use by comparing them with NDBC winds from 22 buoys in the northern Pacific. The mean RMSE was 1.25 m/s and the mean wind angle bias was 10.4°. The spatial and temporal resolutions of the CCMP winds were 0.25°×0.25° and 6 h, respectively.

3.2 Assimilation experiments

Our preliminary experiments included an assimilation run and a control run for each month. The WAVEWATCH III model was hot-started in each run.

To investigate the impacts of the data assimilation on SWH fields in both the assimilation and the hindcasting periods, we designed a 4-day cycle scheme. For example, for a 4-day (96-h) cycle, data assimilation only takes place in the first day (0-24 h) (assimilation period) when Jason-2 passes over the target domain. There is no assimilation over the next 72 h (hindcast period). We used altimeter SWH data within ±0.5 h for each assimilation time. We chose January and July as representative winter and summer months to consider the effect of the SWH assimilation in different seasons. Consider January 2011. There were 63 assimilation tracks over the eight first-days (1, 5, 9, 13, 17, 21, 25, and 29), and 234 over the whole month. In every 4-day cycle, a 24-h assimilation period was followed by a 72-h hindcast period. The output fields were produced at 6-h intervals.

Note that EnOI assumes that only observations located within a certain distance from the given grid point influence the assimilation analysis. This is referred to as localization (local analysis at a grid point using only nearby observations), and avoids problems associated with a large amount of observations. Using the calculated ensemble correlations, we assumed that the average correlation length scale was approximately 500 km, because it ranged between 300 km and 700 km. This suggests a larger influence radius for the local analysis, which we took to be 1 000 km. We used an exponential weight based on the distance between the given point and observations within 1 000 km. That is, grids within 1 000 km of the selected track were affected by the along-track observations. In the first-day of consecutive assimilation, several tracks passed over the Indian Ocean and more than 30% of the grids were affected by the observations. However, these observational influences gradually reduced away from the track, because of the weight of the background innovation ((d - Hψ_f) in Eq.7).

3.3 Along-track SWH data

We used the along-track SWH data from the altimeter products and the final geophysical data records (GDR) of the OSTM/Jason-2 satellite for data assimilation and verification. This data was produced and distributed by Aviso (http://www.aviso.oceanobs. com/) as part of the SSALTO (Segment Sol Multimission Altimetry and Orbitography) ground processing segment. OSTM/Jason-2 revisits the same ground-track within a margin of ±1 km approximately every 9.9 days. The SWH observations from OSTM/Jason-2 were within 0.25 m or 5% accurate, whichever is greater (Dumont et al., 2009). Quality control should be implemented before these observations are used, especially in terms of the horizontal consistency.

4 ENSEMBLE CORRELATIONS

In the EnOI scheme, the background error covariance matrix is estimated by an ensemble error covariance matrix. For this, we sample a stationary ensemble of the model state during a long time integration. When constructing the ensemble, we must ensure that the time series is stationary, with zero mean and no trend. We use an ensemble size of 93, with one anomaly from every month of a 3-year (2008-2010) model spin-up.

4.1 Flow-dependency

To analyze the ensemble correlations, we chose optional reference points P₁₁ (50°E, 3°N) and P₁₂ (92°E, 5°S) for the January ensemble, and P₇₁ (57°E, 13°N) and P₇₂ (80°E, 7°S) for the July ensemble (Fig. 1). Northeasterly winds prevail in January over the north Indian Ocean, so P₁₁ was chosen in the steadily monsoon wind region whereas P₁₂ was chosen for the wind transition zone. We picked P₇₁ and P₇₂ because strong and steady southeasterly winds south of the equator that veer to southwesterly winds north of the equator control the domain in July. Depending on the location of the reference point, the horizontal correlation between the SWH values at the reference point and at all model grid-points indicates that the structure of the ensemble error covariance strongly depends on the downwind monsoon airflow advection being approximately elliptical. From the Somalia jet airflow area to the Arabian Sea, strongly positive correlations between the SWH values at the reference points (P₇₁ for July and P₁₁ for January) and at all model grid-points are distributed along flow patterns of atmospheric forcing. This flow-dependency can also be confirmed by correlations between the forcing wind and wave heights (Fig. 2), indicating that the ensemble error covariance is strongly dependent on monsoon airflow fields. We expect that this flowdependency can be retained in the error covariance analysis (Section 5).

5 RESULT 5.1 Analyses 5.1.1 Along-track test

In our assimilation experiment, data were only assimilated during the first day (24 h) of every 4-day (96-h) cycle when Jason-2 passes over the target domain. Seven passes across the north Indian Ocean and South China Sea (018, 025, 027, 029, 031, 038, and 040) were available for January 1 (Fig. 3, top); and eight passes (081, 088, 090, 092, 094, 101, 103, and 105) were available for July 1 (Fig. 3, bottom). Consider 7°S and 10°S, for example. Figure 4 shows the hourly evolution of the differences in SWH from the assimilation and control runs, for the assimilation time periods on January 1 and July 1. In the bottom figure, the narrow and long belts of negative SWH differences correspond to every track in Fig. 3 (bottom, July 1). Data assimilation pushes the SWH values towards the observations, with the biggest correction more than -0.9 m. Similarly, consider the first day of January and a latitude of 7°S, for example. Note that the crossover point position of Tracks 027 and 040 is approximately on this latitude (about 7°S). Figure 5 (top) shows the situation for January 1. The SWH differences are negative for Tracks 018 and 029, and positive at the crossover points of Tracks 027 and 040, and Tracks 025 and 038. Obviously, the improvements from the data assimilation in January are not as remarkable as in July.

Figure 5 compares the SWH simulated by the model and from the along-track observations (top: January; bottom: July). In these figures, the analysis from the first-day assimilation is denoted by a red triangle and the background (i.e., the first-guess or predicted) with no assimilation is denoted by a black circle. In Fig. 5 (top), the clusters of both background and analysis points are approximately located around the equal line. However, the background estimates are clearly overestimated in Fig. 5 (bottom), especially along Tracks 090 and 101 where the sea-states are dominated by swells (Greenslade, 2001). Tolman (2009) noted that, when comparing model results with observations, the model tends to overestimate swells and has regional biases. This defect is probably due, in part, to errors when generating the non-linear evolution of these swells.

The improvements from the assimilation for July 1 are more remarkable than for January 1. The RMSEs of the control run were up to 0.69 m and 0.84 m along Tracks 090 and 101, respectively; whereas the RMSEs of the assimilation run were 0.19 m and 0.17 m, respectively.

5.1.2 Statistics for the whole month

We ran the model for every other 4-day cycle of the months. There were 63 assimilation tracks for the eight first-days (days 1, 5, 9, 13, 17, 21, 25, and 29). A summary of the validation results averaged over all 63 assimilation tracks is shown in Table 1. The statistics include the correlation coefficient (c.c.), mean bias, and RMSE. The correlation coefficients for January and July 2011 were both 0.98. There was a 40% reduction in RMSE for January, and up to a 65% reduction for July. There was also an astonishing improvement in mean bias for July. The most likely reason for this is that summer monsoon winds are stronger than winter monsoon winds in the north Indian Ocean, resulting in more monsoon-generated swells in July than January. Additionally, in winter, this area receives weaker incoming swells from the Southern Hemisphere when compared with summer, which are appropriately modelled by the two-way nesting scheme.

In summary, these dramatic improvements suggest that altimeter SWH data assimilation using the present EnOI scheme is very effective.

In Section 4, we showed the structure of the ensemble error covariance is significantly flowdependent. We expect that the error covariance of the analysis fields is also flow-dependent. We computed the correlation between SWH at the reference point and at all model grid-points in the target domain. Figures 6 (January) and 7 (July) show the structure of the analysis error covariance. The black dots in Fig. 6 denote P₁₁ (left) and P₁₂ (right), and denote P₇₁ and P₇₂ in Fig. 7. It is clear that significant flow-dependent features are retained in the analysis error covariance. The differences (analysis error covariance minus background error covariance) are negative (see the bottom rows in Fig. 6 and 7), which indicates that the present EnOI-based assimilation is effective.

5.2 Hindcasts

The long revisit period of OSTM/Jason-2 makes it difficult to provide timely measurements of the same or neighboring tracks to verify the hindcasts. For this reason, we used a threefold division of the remaining three hindcasting days (i.e., 0-24 h, 24-48 h and 48-72 h) to consider the trends in the operational hindcasts. We calculated verifications and error statistics, and assessed the impact of the assimilation on the hindcast results by comparing these three segments.

5.2.1 Along-track comparisons

In addition to considering the along-track values, we used the following comparisons in terms of the three time segments. We ran hindcasts that started from the improved initial fields for 0-72 h. The hindcast results were compared along the tracks with the observed SWH data. A summary of the comparison is shown in Table 2. We mainly used the RMSE of the SWH field to assess the overall improvement of hindcasts, because it is a combination of the bias and the standard deviation. We considered the improvements in the SWH RMSEs (assimilation compared with control) according to the three time segments. There was more than a 5% improvement in the 0-24-h hindcast (i.e., at the end of July 2). However, there were no significant improvements in the January 2-4 hindcast, due to an ineffective assimilation in the initialization period.

Note that the large computational domain is such that the limited tracks and along-track data provided by the single satellite (Jason-2) only influence the local region around the track, which leaves large gaps. Therefore, tracks used for verification in the 0-72-h hindcast period must be carefully chosen. It is difficult to effectively assess the results if the selected track does not pass through the influence region of the assimilation, or if it has been a long time since the last assimilation. This may also be misdiagnosed as the assimilation being too short. In other words, we should note that the full impact of the assimilation cannot be fully assessed because of the lack of information in areas outside the satellite ground track.

5.2.2 Comparisons along a latitude line

We also separately compared the 0-72-h hindcast results (based on the assimilated and non-assimilated initial fields) and the observed SWH data with respect to the grid, e.g., along a latitude of the domain. The aim was to investigate the differences between the assimilated and non-assimilated hindcast results, and how long the effect of assimilation extends into the hindcast period.

Consider the assimilated and non-assimilated results for the first four-day period of July, and the results along a latitude of 10°S. Figure 8 shows some obvious differences between the two hindcasts. The negative difference along each track indicates that SWHs were overestimated in the control run, with a maximum difference of 0.5 m. significant differences greater than 0.1 m persisted up to approximately 24 h. However, the differences gradually decreased until disappearing, as the hindcasting time increased.

The most remarkable improvements were for Track 103, which passed though the domain at 2100 UTC July 1. Figure 9 contrasts the mean SWH for Track 103 from the assimilation and control runs (top), and the differences between them (bottom). These results suggest that the assimilation impact lasts for no more than 48 h.

6 DISCUSSION AND CONCLUSION

Our assimilation experiments suggest that we should consider the effect of SWH assimilation in different regions of the Indian Ocean and different seasons. We conclude that the SWH data assimilation had the greatest impact when the sea-states are dominated by swells.

The clusters of background and analysis points for January and July are shown in Fig. 5. We analyzed these along-track comparisons and contrasted them with the ground tracks (according to track-number and passing time), as shown in Fig. 3. For January, the assimilated model reduced the RMSEs of Tracks 018, 029, 040, 027, 025, and 038 by 22.2%, 30.0%, 28.6%, 42.3%, 51%, and 59.6, respectively. Similarly, for July, the RMSEs of Tracks 094, 105, 103, 090, 090 and 101 were reduced by 60.0%, 53%, 67.2%, 71.2%, 72.5%, and 80.0%, respectively. These track numbers are listed from west to east. The assimilation had the greatest impact at eastern locations (pointing toward Java and off the coast of northwest Australia), where the sea-state is dominated by swells (Greenslade, 2001) in January and July. However, there were smaller RMSE reductions in January than in July. This is essentially in agreement with one of the conclusions in Greenslade (2001), which was that the assimilation (using statistical interpolation) had the greatest impact around the west coast of Australia, where the sea-state is dominated by swell.

July is in the summer monsoon period of this area, when steady and strong prevailing winds are sufficiently long to promote matured wind-sea fetches. At the same time, a great extent of the lower latitudes of the domain is dominated by swells that mainly originate in the southern hemisphere. July is in the southern hemisphere’s winter, when rough weather and stronger storms result in much more outward-going swells, which propagate thous and s of kilometers to the target domain through open boundaries. This is appropriately modelled by the two-way nesting scheme. In contrast, in January, the northern Indian Ocean is not extensively subjected to strong swells, and instead there are weak prevailing winds. In conclusion, the SWH data assimilation had the greatest impact when and where sea-states were dominated by swells.

One of the main innovations of this work is our development of the wave data assimilation system based on the EnOI. The EnOI-based assimilation system for wave hindcast experiments in the north Indian Ocean area produced preliminary and satisfactory results. This study is a first step towards an EnOI-based data assimilation wave system, which, in the long run, will provide a basis for operational applications. We have presented the advantages of the EnOI and the sampling strategy for its stationary ensemble. The performance in terms of the altimeter SWH data assimilation was assessed by comparing different experiments and observations.

The EnOI assimilation technique uses an estimate of a spatial error covariance structure based on a stationary ensemble of samples, so it is important that the ensemble members are chosen appropriately. Our approach is based on a long time model integration, picking daily mean SWH fields from the same month of different years. The update process for the sampling strategy of the ensemble in monsoon areas must be based on the month in question. The significant flowdependent BEC distribution shows that this sampling approach is reasonable.

To investigate the impact of the EnOI-based assimilation of altimeter wave height data on the analysis fields and 0-72-h hindcasts in different seasons, we ran assimilation and control experiments for January and July 2011. When compared with altimeter SWH observations and results from the control run, the EnOI-based assimilation results were significantly better in terms of the SWH RMSEs of both the analyses and hindcasts.

There was a considerable systematic bias in the control SWH analyses for July 2011, and it was much larger than for January. However, a monthlong consecutive assimilation of Jason-2 SWH data reduced the systematic bias in July 2011 by approximately 84% and the RMSE by approximately 65%. The initial field for the four-day-cycle scheme improved significantly, and was then used for the 0-72-h hindcast. This provided a better estimate of the true sea states. Improvements in the SWH values were more remarkable in July than in January.

The present EnOI-based wave assimilation system is promising for operational applications because of its low computational cost. For operational forecasts, the forcing winds should be real-time model forecast winds instead of historical winds.

ACKNOWLEDGEMENT

The authors wish to thank Professor ZHU Jiang and Dr. XIE Jiping at the Institute of Atmospheric Physics, Chinese Academy of Sciences for their valuable discussions and suggestions in the initial stages of this study.