پیش بینی و جذب پردازش منطقه گشت و گذار با استفاده از شبکه های بیزی: قسمت دوم: مدل معکوس

A Bayesian network model has been developed to simulate a relatively simple problem of wave propagation in the surf zone (detailed in Part I). Here, we demonstrate that this Bayesian model can provide both inverse modeling and data-assimilation solutions for predicting offshore wave heights and depth estimates given limited wave-height and depth information from an onshore location. The inverse method is extended to allow data assimilation using observational inputs that are not compatible with deterministic solutions of the problem. These inputs include sand bar positions (instead of bathymetry) and estimates of the intensity of wave breaking (instead of wave-height observations). Our results indicate that wave breaking information is essential to reduce prediction errors. In many practical situations, this information could be provided from a shore-based observer or from remote-sensing systems. We show that various combinations of the assimilated inputs significantly reduce the uncertainty in the estimates of water depths and wave heights in the model domain. Application of the Bayesian network model to new field data demonstrated significant predictive skill (R2 = 0.7) for the inverse estimate of a month-long time series of offshore wave heights. The Bayesian inverse results include uncertainty estimates that were shown to be most accurate when given uncertainty in the inputs (e.g., depth and tuning parameters). Furthermore, the inverse modeling was extended to directly estimate tuning parameters associated with the underlying wave-process model. The inverse estimates of the model parameters not only showed an offshore wave height dependence consistent with results of previous studies but the uncertainty estimates of the tuning parameters also explain previously reported variations in the model parameters.

Large spatial and temporal variability in waves, water levels, currents, and bathymetry characterizes the nearshore coastal environment. These variables are typically strongly coupled, as can be illustrated using a variety of numerical models. Therefore, accurate predictions of any of these variables require accurate measurements of boundary-condition data, including details of the incident wave spectrum, water level variations, and bathymetry. Numerous studies demonstrate significant model prediction skill when accurate data are available, e.g., SWAN (Simulating Waves Nearshore, Booij et al., 1999 and Ris et al., 1999), Delft-3D (Lesser et al., 2004 and Reniers et al., 2007), or ADCIRC (Westerink et al., 2008). A modern challenge that must be overcome to use these advanced models for both practical applications as well as for scientific study is to obtain appropriate initial and boundary data from often sparse, noisy, or disparate data sources. In essence, the model capabilities often exceed the quality of the data used to force the models. Numerous approaches to this problem have been implemented using, for instance, spatial interpolation (Plant et al., 2009) or formal data-assimilation methods (e.g., Feddersen et al., 2004). Researchers of global ocean circulation and weather have recognized that data assimilation is a required component of their research and operations (e.g., Goerss, 2009). This recognition has been slower among those studying the shallow regions near the coastline. Here, we demonstrate a new methodology that is appropriate for assimilating data and models that are available in the nearshore environment. In our companion paper (Plant and Holland, 2011), we presented a Bayesian network approach for making wave-height predictions and associated prediction errors across the surf zone where the network acted as a forward model. The Bayesian network approximates the joint probability between system variables (e.g., wave height, period, direction, and water depth) that were expected to be correlated. A specific system variable can be estimated using constraints on related variables provided by observations or other data. The approach assumed that both the model and the data were potentially inaccurate. Specifically, data inaccuracies (including measurement errors and spatial under-sampling) and model inaccuracies were captured in parameter errors. The Bayesian network was trained through the assimilation of realistic simulations provided by a 1-dimensional (cross-shore) deterministic numerical model. When new data inputs and input errors were supplied to make a prediction, the Bayesian approach made skillful predictions of both measured wave heights and prediction uncertainty. Another capability of the Bayesian network approach, not described in the companion paper, is that it may be applied in an inverse sense. That is, it can be used to efficiently assimilate observational data (e.g., wave heights) in the interior of a model domain in order to update knowledge of the boundary conditions. The utility of this sort of assimilation is threefold. First, updated boundary conditions might be intrinsically useful. For instance, assimilation methods have been used to estimate nearshore bathymetry (Piotrowski and Dugan, 2002, Plant et al., 2008 and Stockdon and Holman, 2000), which could be used independently of the network for navigation, safety, or validation of morphologic evolution models. Also, more highly resolved or accurate boundary conditions estimated using the Bayesian approach could be used to drive related, more detailed numerical models. Second, since the updated information in a Bayesian network can propagate to both boundary conditions and to variables within the model interior, observations can be used to optimally update all modeled variables simultaneously. The Bayesian network will make a prediction that appropriately weights the new information with respect to its prior predictions. Thus, if new data are very accurate, the Bayesian prediction will match all the data; whereas, if the data are very inaccurate or inconsistent, the prediction will be unaffected by the assimilated data. Third, the Bayesian network can be extended to assimilate variables that are not typically used as either input or output within detailed, numerical nearshore process models. For instance, observations of sandbar positions are available at some coastal locations (Lippmann and Holman, 1989, Plant et al., 1999 and Ruessink et al., 2003b). Sandbars certainly indicate something about the bathymetry (it is shallow at the bar crest) and can have a direct impact on wave energy dissipation, but since the numerical depth of the bar is not known, sandbar position cannot be used directly as a numerical wave model input. In contrast, assimilation of sandbar position into the Bayesian network is straightforward. Other examples of observations not typically assimilated include surf zone width and the intensity of wave breaking. In this paper, we investigate the inverse and assimilation capability of a Bayesian network developed from the 1-dimensional (1-D) process model described in detail in the companion paper. In Section 2 (this paper, Models), we briefly describe the previous work and introduce an extension that includes three additional variables: inner and outer sandbar positions and the normalized wave height (i.e., the local ratio of wave height to water depth, which is related to the intensity of wave breaking). These variables are selected because they can be estimated via remote sensing. In Section 3 (this paper, Applications), we test the ability of both the original and the extended Bayesian networks to estimate boundary conditions and assimilate data. As with the forward modeling (companion paper), we test both the ability to make accurate predictions and to estimate prediction uncertainty. Discussion of the implication of the analysis results and of sensitivity to input errors is presented in Section 4. Conclusions are presented in Section 5.

A Bayesian network model has been developed to simulate a relatively simple problem of wave propagation in the surf zone. In the companion paper (Part I), we showed that the Bayesian approach captured the interactions predicted using a detailed numerical model and accurately propagated uncertainty from the input to the output. Here, we demonstrated that this model can provide both inverse modeling and data-assimilation solutions under reasonably complex scenarios. In the inverse modeling case, we demonstrate that offshore boundary conditions can be efficiently estimated from very limited data input in the interior of the model domain. Specifically, a single observation of water depth and wave height (or even wave height alone) near the shore could be used to make a useful (i.e., better than wave-height climatology) prediction of the offshore conditions. This approach was tested against an independent data set that was not used to train the network in order to predict offshore wave heights. Best results yielded mean prediction errors of about 10 cm (rms error of 30 cm, and a correlation skill of about 0.7). Best results were obtained by allowing information about offshore water depths to be omitted (but this was not a general result, and was related to an uncommon bathymetric configuration). Input uncertainties (including exaggerated uncertainty on the depth values and limited uncertainty in parameter values) allowed the Bayesian network to correct input errors and other errors introduced by limitations in the methodology. The inverse solution was extended to allow data assimilation using observational inputs that were not compatible with detailed numerical implementations of the problem. These inputs included sandbar positions instead of bathymetry and estimates of the intensity of wave breaking instead of wave height observations. The sandbar-position data served to constrain the bathymetry more effectively than a single direct observation of water depth. Information about wave breaking was shown to be essential to reduce prediction errors, with the best estimates of the offshore wave height resulting when the normalized wave height was known at both the landward and intermediate locations. These results strongly indicate that visual observations and remote-sensing data can be used very effectively to reduce prediction uncertainty. Application of the model to a field example demonstrated significant predictive skill (R2 = 0.7) and enormous improvement over an alternative climatological prediction.