High-quality wave prediction with a numerical wave model is of societal value. To initialize the wave model, wave data assimilation (WDA) is necessary to combine the model and observations. Due to imperfect numerical schemes and approximated physical processes, a wave model is always biased in relation to the real world. In this study, two assimilation systems are first developed using two nearly independent wave models; then, “perfect” and “biased” assimilation frameworks based on the two assimilation systems are designed to reveal the uncertainties of WDA. A series of biased assimilation experiments is conducted to systematically examine the adverse impact of model bias on WDA. A statistical approach based on the results from multiple assimilation systems is explored to carry out bias correction, by which the final wave analysis is significantly improved with the merits of individual assimilation systems. The framework with multiple assimilation systems provides an effective platform to improve wave analyses and predictions and help identify model deficits, thereby improving the model.

Ocean waves, referring to the ocean surface gravity waves driven by wind, are important physical processes in the study of multiscale coupled systems. Many studies show that ocean waves are necessary for upper-ocean mixing processes, whether in small-scale coastal simulations or large-scale global climate simulations (e.g., Babanin et al., 2009; Huang and Qiao, 2010; Qiao et al., 2004, 2010). The existence of ocean waves can modify the structures of both atmospheric and marine boundary layers by providing sea surface roughness, wave-induced bottom stress, breaking-wave-induced mixing and so on, which ultimately influence air–sea momentum and heat exchange. Therefore, ocean waves are an important component in atmosphere–ocean interaction flux processes (e.g., Chen et al., 2007; Doyle, 2002; Liu et al., 2011; Warner et al., 2010). In addition, the study of ocean waves can reduce and prevent marine disasters and provide guidance for the development of the social economy (e.g., Folley and Whittaker, 2009; Rusu, 2015; Wei et al., 2017). Thus, studying ocean waves is of great scientific and social significance.

At present, ocean wave observational techniques are constantly being improved (e.g., Daniel et al., 2011; Hisaki, 2005). Except for traditional buoy observations (e.g., Mitsuyasu et al., 1980; Rapizo et al., 2015; Walsh et al., 1989), satellites can provide much near-real-time wave observational information, which is beneficial for understanding the state of ocean waves (e.g., Gommenginger et al., 2003; Lzaguirre et al., 2011; Queffeulou, 2004). However, observations always represent scattered samples in time and space in the real world and therefore do not represent the complete three-dimensional structure and temporal evolution of real-world waves.

Numerical wave models are a powerful tool for studying the physical processes of ocean waves and predicting future wave states. Following the development of the previous two generations, third-generation wave models, such as WAve Modeling (WAM) (WAMDI Group, 1988), WaveWatch III (WW3) (Tolman, 1991), Simulating Waves Nearshore (SWAN) (Booij et al., 1999), and MArine Science and Numerical Modeling (MASNUM) (Yang et al., 2005), integrate the spectral action balance equation describing the two-dimensional ocean wave spectrum evolution without additional ad hoc assumptions regarding the spectral shape, and these third-generation models are more robust for arbitrary wind fields than previous models. However, there are generally three error sources in wave models. One error source is from an incomplete understanding of the physical processes, approximate expressions of the numerical discretization schemes and so on, which causes systematic errors that are usually referred to as wave model bias. The second error source is due to inaccurate wind forcings of wave models. The third error source is from the initial-condition uncertainties, which can grow due to the nonlinearity of the model equations during model forwarding. In this sense, the model-simulated waves do not represent the real world either.

Given the scattering nature of observational information and the approximate characteristics of wave modeling, wave model data assimilation (WDA) is necessary to combine the advantages of both the model and observations. WDA optimizes the model initial conditions to produce more accurate wave forecasts and produces a more accurate evolution of three-dimensional wave states to elucidate the underlying mechanisms; this approach dates back to the 1980s (e.g., Esteva, 1988; Janssen et al., 1989). Since then, many advanced WDA methods have been developed (e.g., Abdalla et al., 2013; Bauer et al., 1996; Greenslade and Young, 2004; Jesus and Cavaleri, 2015; Lionello et al., 1992; Sun et al., 2017; Vorrips et al., 1999), and their applications have been assessed (e.g., Francis and Stratton, 1990; Heras et al., 1994; Stopa and Cheung, 2014). Furthermore, various observation types, such as buoy, radar and satellite, have been applied to WDA (e.g., Bhatt et al., 2005; Breivik et al., 1998; Feng et al., 2006; Greenslade, 2001; Hasselmann et al., 1997; Qi and Cao, 2016; Voorrips, 1999; Waters et al., 2013), and the wave forecasts have also been directly addressed (e.g., Almeida et al., 2016; Emmanouil et al., 2012; Lionello et al., 1995; Qi and Fan, 2013; Sannasiraj et al., 2006; Voorrips, 1999; Wang and Yu, 2009; Zhang et al., 2003).

Due to the approximate nature of the numerical discretization and physical processes, a systematic difference between a model and the real world (i.e., model bias) exists. As noted by Zhang et al. (2012), since model bias is not well defined in observational space, the influence of model bias on data assimilation is a challenging research topic. Alternatively, one can simulate model bias using a pair of models and study the adverse impacts on data assimilation. Inspired by previous work (e.g., Dee, 2005; Zhang et al., 2012), here, we use a simple data assimilation scheme with two wave models (WW3 and SWAN) to explore the influences of different error sources on WDA. The adverse impacts of wind forcing errors and initial-condition uncertainties as well as wave model bias on WDA are studied first, and then two simple statistical methods for bias correction are developed to mitigate assimilation errors and improve wave analysis.

This paper is organized as follows. After the introduction, the methodology is presented in Sect. 2, including a brief description of the employed models and observations, the development of the two WDA systems using the WW3 and SWAN models, as well as the design of experiments throughout the study. Section 3 presents the model bias analysis and the adverse impacts of model bias on WDA. In Sect. 4, the method used to mitigate model bias influences on wave assimilation is explored. Finally, the discussion and conclusion are given in Sect. 5.

In the wave models, the variance spectrum or energy density

In this study, we use three advanced third-generation spectrum models – WAM (Cycle 4.5.4) (WAMDI Group, 1988), WaveWatch III (WW3, version 5.16) (Tolman, 1991) and SWAN (version 41.20) (Booij et al., 1999) – which have improved a lot from their predecessors, such as numerical and physical approaches. These wave models have the same form of governing equation; however their numerical implement processes are different due to different considerations. For example, SWAN is more focused on wave propagation processes in shallow water, while WAM and WW3 pay more attention to deep water. For more comparisons, please see Sect. 2.2.

Three wave models use two-dimensional spectral space containing 29 frequencies that cover from 0.035 to 0.555

The AVISO (Archiving, Validation and Interpolation of Satellite Oceanographic) data (

The ground projection tracks of satellite Jason-2 (blue line, with
an inclination of 66

Since the observations are only a sample of real-world information, the model bias (i.e., systematic difference between a numerical wave model and the real world) is not well defined against the real world. In this study, we use the systematic difference between the WW3 and SWAN models to simulate the model bias and study the influences on wave data assimilation (WDA).

First, let us identity the difference in physical and numerical aspects to comprehend the causes of “bias” between these two models. In general, WW3 addresses global scales, and SWAN is more applicable in shallow water. Although the two models have most of the same physical processes, such as the wind input and nonlinear wave–wave interactions, each can provide multiple parameterization schemes to choose. For example, the nonlinear wave–wave interactions in SWAN include the discrete interaction approximation (DIA) (Hasselmann et al., 1985) and the Webb–Resio–Tracy (Resio and Perrie, 1991; Van Vledder, 2006; Webb, 1978), while there are more choices in WW3, such as the Generalized Multiple DIA (Toman, 2004, 2013), the two-scale approximation and full Boltzmann integral (Perrie et al., 2013; Perrie and Resio, 2009; Resio et al., 2011; Resio and Perrie, 2008), as well as the nonlinear filter scheme (Tolman, 2011). In order to reduce more uncertainties, the same scheme is chosen for the same physical process. In numerical aspects, there exist different implementation strategies such as the differencing method, which also contributes to bias.

To explore the model bias influences on WDA, we develop two data assimilation systems based on WW3 and SWAN in this study.

Generally, based on the program structure of wave models, we insert the assimilation module between calculations of the two-dimensional wave spectrum and outputs of wave parameters so that at the assimilation time, we call on the assimilation module to update the spectrum and SWH. When building the data assimilation systems, we need to consider the different structures of parallelism method, data storage and information exchange in WW3 and SWAN models as noted in Sect. 2.2.

To clearly demonstrate the influences of model bias on WDA and minimize its adverse impact, the analysis scheme in both assimilation systems is optimal interpolation (OI), which also is low cost and easy to operate. We implement the OI analysis in three to four steps. The first step uses two Gaussian convolutions of the background and observed SWHs to compute the observational increment of SWH at the observational location. The second step projects the SWH observational increment onto the model grids centered at the observational location but within an impact radius using linear regression. The third step transforms the analyzed SWH to update the spectrum of model waves. The fourth step corrects wind forcing using the observational SWH.

Starting from the idea of the ensemble adjustment Kalman filter (Anderson, 2001), an observational increment at the observational location

The second step projects the observational increment

Spatial distributions of

The assimilation SWH

If the assimilation only adjusts the wave spectrum as described in Step 3, the updated spectral structure may be quickly overwritten by erroneous
wind. In this step, we describe a simple scheme using the observed SWH data to correct the wind forcing. Starting from a first guess of wind (the
ERA-Interim reanalysis, for instance), the analyzed wind

With respect to the configuration of wave model data assimilation, the model time step is 15

Throughout this study, we use the symbol MA

Twin experiments refer to a type of observing system simulation experiment (OSSE), in which a model simulation is used to define the “true” solution of a data assimilation problem and the other model simulation is used to start the assimilation. The “observations” are samples of the truth with some white noise to simulate the observational errors. When the truth and assimilation are conducted by different (or identical) models, the framework is a “biased” (or “perfect”) model twin experiment. Within a twin experiment framework, any aspect of assimilation skills can be measured as the degree to which the truth is recovered through the assimilation.

In a perfect twin experiment, we assume that the assimilation model and the observation are unbiased; i.e., both the instrument measuring and numerical modeling processes are sampling the same stochastic dynamical system. Such sampling only has random sampling errors without any systematic difference (bias). We can build this perfect model framework by using the same model to produce the truth as the assimilation model but with different initial conditions and wind forcings.

The observations from the observational time window (1

To study the impact of model errors on wave assimilation, we use two models to design a biased twin experiment. Again, due to the scattering
nature of the observations, it is difficult to obtain a complete picture of the model bias against the real world. Given the difference between the
WW3 and SWAN models described in Sect. 2.2, we use these two models and their assimilation systems here to simulate the model bias and examine its
influences on the WDA. We use the ERA-Interim reanalysis wind to force the WW3 (or SWAN) to produce the truth and observations but use the
SWAN (or WW3) assimilation system to assimilate the observations. The degree to which the truth produced by different model-based
assimilation systems is recovered by assimilating the observations is an assessment of the model bias influences on the WDA. Such a biased
twin experiment can be named WW3

Under the biased twin experiment framework, we also conduct experiments to examine the impacts of observing systems on wave assimilations by
increasing the observational information based on multiple satellite tracks. For example, we can examine the assimilation results of
WW3

In this study, we also conduct real-data assimilation experiments using WW3 and SWAN assimilation systems with real track data from Jason-2,
Jason-3 and SARAL. Through real-data assimilation experiments with different model-based assimilation systems, we can (1) increase our
understanding of the influences of model errors on the WDA and (2) study the method to reduce the model error influences on the assimilation
results. The real-data assimilation experiments can be directly named, i.e., WW3

List of perfect model twin experiments.

Usually, wave numerical simulation can be improved by three methods: (1) reducing the errors in the initial conditions, (2) enhancing the accuracy of the wind forcing, and (3) improving the representation of the wave model and its parameterization.

In this section, we use perfect model twin experiments (as described in Sect. 2.4.1) to exclude model errors and explore the impact of wind forcings
and initial conditions on the wave simulations. To compare the performances of the WW3 and SWAN models, we conduct separate experiments with these two
models. The truth and model control runs are two basic experiments of the perfect twin experiment framework. We use the ERA-Interim wind to drive
WW3 (or SWAN) and generate a long time series of model states as the “truth,” which is called WW3

First, we conduct two sets of model control experiments WW3

The time series of RMSEs of the

The SWH RMSE is approximately 0.34

In this section, using the same model states (at the 45th day) in the corresponding model control as in Sect. 3.1 as the initial conditions, we
conduct two sets of assimilation experiments: WW3

From Fig. 3, we can see that in both models, the assimilation errors are reduced when more observational information is used. The corresponding RMSE reductions in these three experiments from the model control run are 24 %, 32 % and 38 % for WW3 and 26 %, 35 % and 38 % for SWAN, respectively. However, when more satellite observations are assimilated into the model, the magnitude of improvement becomes small (further reduced by 8 % in WW3 and 9 % in SWAN when Jason-3 is added as well as only a 6 % in WW3 and 3 % in SWAN further reduction when SARAL is further added). These results suggest that given wind forcing errors, increasing observational information can help to improve the model behavior, but the improvement is limited.

As described in Sect. 2.2, the WW3 and SWAN models discretize the wave-action-governing equation with different physical processes, parameterization
schemes and differencing schemes. These differences result in each wave model having its own distinguished characteristics. To study the adverse
impact of the model bias on the wave assimilation, the biased twin experiments described in Sect. 2.4.1 are used in this section, where the truth
model and the assimilation model are different between WW3 and SWAN. For example, the WW3

List of biased model twin experiments.

The RMSEs and correlation coefficients produced by the all biased model assimilation experiments are plotted in Fig. 4. The black line in each panel
represents the result of the WW3 model control forced by the ERA-Interim wind (WW3

The time series of RMSEs

Compared with the model controls WW3

The results of two other sets of assimilation experiments called WW3

These assimilation results clearly show that even though the wind forcing is perfect, once a biased assimilation model is used, the wave simulation has large errors. Although WDA can greatly reduce the simulation error by assimilating the observational information into the model, due to the existence of the model bias, the error remains at some significant level and cannot be eliminated entirely even by increasing the observational constraints through an improvement in the observational system and constraint of the wind forcing.

Comparing the time series of SWH RMSE caused by two important error sources, model bias (0.58

Distributions of SWH RMSEs caused by wind forcing

A spatial pattern of SWH RMSE is also displayed in Fig. 5. (Due to the similar performance inside twin experiments, here we only show the results using
WW3 model as simulation/assimilation model.) In the model control run, it makes sense that the error caused by wind forcing (panel
a (WW3

It is worth mentioning that there is a similar performance about the effect of the observation system on improving both error sources, whether in time series or spatial distribution. The more observation information is absorbed into assimilation systems, the better the error improvement in both twin experiments. However, due to the existence of model bias, this improvement has a limitation and stays at a certain level. If more powerful observation is absorbed (such as wave direction, wave period and two-dimensional wave spectra), the limitation maybe stay at a lower level. Next, with the results of real-data assimilation where both the model and wind forcing have errors, we will analyze and discuss how to mitigate the model bias influences on the WDA.

Distributions of SWH mean errors (against the merged grid altimeter data) of the WW3

From the above analyses of twin experiment results, we learned that the model bias has a strong adverse impact on WDA. To explore the method of
mitigating the model bias influence on the WDA, we conduct the real-data assimilation experiments (same time range as the twin experiments)
described in Sect. 2.4.2 using the WW3 and SWAN assimilation systems to assimilate the track data of Jason-2 and Jason-2

Comparing panel a with panel d in Fig. 6 reveals that a large difference exists in the simulations of the two models. First, the SWAN simulation
errors are generally larger than the WW3 simulation errors. Second, the global error distributions are quite different: while the WW3 simulation
errors appear negative (or positive) over most of the 30

The above systematic differences between the two model simulations have significant influences on the results of the WDA. In general, the distribution
of assimilation errors shares the same patterns as the model simulation errors but with a much smaller magnitude. The net result is that both the WW3
negative (or positive) error magnitude over the 30

Time series of RMSEs

Figure 7 displays the time series of the RMSEs and spatial correlation coefficients with the global statistics in space. The RMSE (or correlation
coefficient) of the SWAN model simulation is larger (or smaller) than the WW3 model simulation (0.66

Time series of

The mitigation of model bias is a complex issue in which improving the model is a final but long-lasting solution. From Fig. 6, we learned that the WW3 and SWAN assimilation errors have some common (or opposite) characteristics in some locations. For example, while the SWH over the southern ocean coastal area always appears to be overestimated because of the lack of adequate observations to improve in both assimilation systems, the WW3 and SWAN assimilations appear to be the opposite in the Antarctic Circumpolar Current area and the tropical oceans. The WW3 (or SWAN) assimilation errors in the Antarctic Circumpolar Current area appear positive (or negative), while the WW3 (or SWAN) assimilation errors in the tropical oceans appear negative (or positive). A question arises: as the first step of mitigating model bias influences on the WDA, can we use a pair of assimilation systems to explore a statistical approach to reduce the wave assimilation errors?

Given the opposite behaviors of two assimilation systems existing in certain places, the simplest bias correction could be conducted by a simple average. This assumes that each wave model assimilation system has its own characteristics of systemic error due to deficit physics, and these systemic errors (or model biases) from different wave models follow a Gaussian distribution with a trivial expectation. The corresponding results are shown in Fig. 8. Compared with the performance of each individual assimilation system (dashed red lines for WW3 and dotted red lines for SWAN), the results of this bias correction (cyan lines) show that the RMSE is reduced (Fig. 8a) but the spatial correlation is not greatly improved (Fig. 8b). It is reasonable that based on the opposite errors deviating from the real world in two assimilation systems, this correction method employing the mathematical average can reduce the RMSE to some extent, but it may not have a significant contribution to improving the correlation coefficient if either the sampling size of model bias is too small (only two in this case) or the bias has an asymmetric distribution.

Distributions of corrected SWH RMSEs of the WW3

Considering the potential asymmetry of the Gaussian distribution of different model biases and small sampling size in practice, as the first step, we
calculate the spatial distribution of model bias in every assimilation system (the time mean of the difference between observations and assimilation
results) (Fig. 6) and extract it at each time step in the output and then calculate the expectation (average) of all assimilation systems as the
results after bias correction. The corresponding results are shown with pink lines in Fig. 8. Both the RMSE and correlation are improved greatly. We
also show the spatial distribution of SWH RMSEs after bias correction with the second method. From Fig. 9, we can easily find that after bias
correction, all the experiments have similar error performances, the smaller at low latitudes and the larger at high latitudes. As the satellite observation increases (one in panels b and f and three in panels c and g), the error decreases gradually compared to the model control run (panels a, e). Based on
the best assimilation results currently (panels c, g), the improvement with a corrected wind is more efficient in WW3 (panel d), especially at high latitudes which is the higher error area, but it is hardly to found in SWAN (panel h). If we focus on the change without (panel c) and with (panel d) the
corrected wind in the WW3 model, the RMSE improvement of wind sea and swell is 27 % and 1 % averaged globally. If we divide the global area into the three
parts mentioned in Sect. 3.4, the wind sea (or swell) improvement of the regional RMSE mean is 4 % (or 3 %), 3 % (or 1 %) and
6 % (or 1 %) in the northern westerly zone (65–30

To verify the feasibility and applicability of the bias correction method above, three well-known wave models (WW3, SWAN and WAM) with the same data assimilation method are used to conduct longer assimilation and bias correction experiments. The calculation period lasts for 14 months (from November 2016 to December 2017) with sufficient spin-up process to reach a steady assimilation state (the first month for model spin-up and the second month for assimilation spin-up). The results of the last 12 months (for 2017) are analyzed and presented in Figs. 10 and 11 for the spatial distributions and time series of RMSEs and spatial correlation coefficients, respectively. From Fig. 10, we can see that both the RMSE and the correlation coefficient (panels d and h, respectively) have been improved by the bias correction that combines the advantages of every WDA system (panels a and e for WW3, panels b and f for SWAN, panels c and g for WAM). In Fig. 11, the bias correction of model control runs (the second bias correction method is conducted based on the result of model control run from three wave models) shows improvement but is worse than the data assimilation before bias correction (compare green with pink). Compared with the model control (blue), the assimilation results with bias correction (red) can reduce the error by 25 % and significantly enhance the correlation coefficient (from 0.88 to 0.923). This result confirms that this bias correction based on multiple assimilation systems can effectively enhance the WDA quality.

Distributions of SWH RMSEs

The time series of

Ocean waves cause the sea surface roughness to impact the boundary conditions of the atmosphere and the wind stress of the ocean surface. Wave processes, such as wave breaking and wave-induced bottom stress, have significant effects on ocean mixing. Thus, ocean waves are important physical processes for understanding ocean mixing and air–sea interactions in coupled Earth systems. More accurately predicting ocean waves is of great societal significance. However, multiple error sources exist in wave simulations and predictions, including modeling errors, wind forcing errors and initial-condition errors.

To sort out the source of the errors of wave data assimilation (WDA), a pair of independent WDA systems is first developed using two wave models: Wave Watch III (WW3) and Simulating WAves Nearshore (SWAN). The perfect and biased model twin experiment frameworks are designed to clearly identify each error source and examine its influences on WDA. The results show that model bias is a significant error source that has a largely adverse impact. Then, two WDA systems are used to design bias correction approaches to mitigate the influences of model bias and improve the assimilation quality. Finally, long-term WDA experiments added by the third WDA system with the WAM model (WAve Modeling) (WW3, SWAN and WAM) are conducted to validate the bias correction method. Three findings are established. (1) When the model is perfect, the initial-condition error decays within 10 d, but the WDA can shorten the timescale by half. (2) When the model is biased, despite a perfect wind forcing, the wave simulation has large errors and the WDA can only reduce the error to a limited extent. (3) With the results from two assimilation systems, a statistical approach of bias correction significantly improves the quality of final wave analysis by combining the merits from individual assimilation systems.

Model bias is an obstacle to improving WDA and wave predictions. Using multiple assimilation systems to study the influences of model bias on WDA is an effective approach. As the first step, however, we have used a simple assimilation scheme and simple bias correction method. In follow-up studies, we shall consider powerful observation information (such as two-dimensional wave spectra), advanced assimilation schemes (such as ensemble Kalman Filter) and more comprehensive correction methods to help improve modeling. For example, the “online” bias correction (the “offline” bias correction is used in this paper) during the assimilation process (e.g., Dee, 2005) will be considered to improve the assimilation results and correct the instantons initial condition within individual assimilation systems; after that, a robust observation (such as a buoy) is needed to validate the quality of bias correction. In addition, improving the model is an important, inevitable and long-lasting task. In this study, under the same wind conditions, we find that three models show common bias characteristics in the Antarctic Circumpolar Current (ACC) area. If there is a similar performance forcing by other wind, this may suggest that present wave modeling may have deficits in energy spectrum expression for high-wind-speed areas. In the future, we will further examine the sensitivities of physical processes on high wind speed to mitigate such a common modeling bias. All in all, a robust bias correction method with a lower model bias and higher representation of wave physical characteristics may further improve wave analysis quality. Once a long time series of high-quality wave analyses is available, it is expected that we can improve our understanding of ocean mixing. The physical process of wave-induced mixing is linked with the structure of the ocean mixing layer (Qiao et al., 2010). This process can be expressed as a function of wave number, frequency and wave spectrum and so on, provided by wave analysis. With the framework of multiple WDA systems developed in this study, improved wave predictions can be effectively pursued. How can we further enhance the predictability of ocean waves? The first important step is to understand the physical process of ocean waves better based on a more accurate evolution of wave state from this framework. Answering these questions could be very important and interesting research topics for future studies.

Codes, data and scripts used to run the models and produce the figures in this work are available on the Zenodo site (

JL coded the module of data assimilation and inserted it into the wave models and carried out all the experiments. SZ designed the experiments and analyzed the results with constructive discussions. Both authors wrote the paper together.

The authors declare that they have no conflict of interest.

The research is supported by the National Key R&D Program of China (grant nos. 2017YFC1404100 and 2017YFC1404102), the National Natural Science Foundation of China (grant nos. 41775100, 41830964) and the Fundamental Research Funds for the Central Universities (grant no. 201961001).

This paper was edited by Qiang Wang and reviewed by two anonymous referees.