Automatic calibration and uncertainty quantification in waves dynamical downscaling
Introduction
Waves are responsible for the highest loads on coastal structures, are the main forcing of sediment transport in beaches and condition any activity developed on coastal waters. Therefore, long-term and good quality wave data series are essential for any coastal engineering project. At the same time, as probabilistic approaches become standard practice in many coastal engineering applications, it is not enough to obtain reliable long-term wave data series, but it is also necessary to have an estimation of their uncertainty.
Despite its relevance, reliable coastal wave data is only available in some areas, where long-term wave measurements are available and have been used for calibration and validation of high-quality and high-resolution wave hindcasts (e.g. O'Reilly et al., 2016 for the coast of California; Shimura and Mori, 2019, for Japan). For most regions, typically in developing countries, engineers commonly resort to global wave hindcasts (e.g. ERA 5, Hersbach et al., 2020), ideally in combination with a short-term (i.e. a few months) in-situ measured series obtained for the particular project with the aim of calibration or validation of the hindcasts.
The most recent global wave hindcasts (Chawla et al., 2013; Rascle and Ardhuin, 2013; Perez et al., 2017; Hersbach et al., 2020) provide more accurate results on coastal regions than its predecessors that were focus exclusively on oceanic deep waters (e.g. Cox and Swail 2001; Sterl and Caires 2005; Reguero et al., 2012). However, the resolution of the coastal grids is still not fine enough to capture small-scale variability in the nearshore, they do not consider local bathymetric information and are validated with or assimilates almost exclusively deep waters altimetry data. Therefore, recent global wave hindcasts provide better quality and closer to site boundary conditions but they still must be downscaled to be used on a coastal engineering project (Perez et al., 2017).
Downscaling can be done with statistical methods (e.g. Hegermiller et al., 2017; Camus et al., 2014) or, more commonly, through the use of numerical models that solve the set of equations that mathematically model the physics of the waves propagating to the coast (e.g. Rusu et al., 2008), known as physical or dynamical downscaling. Methodologies that combine dynamical and statistical downscaling techniques are called hybrid downscaling (e.g. Camus et al., 2011a). In recent years downscaling of met-ocean variables to the coast has received considerable attention; e.g. many of the works carried out in the framework of the European project CEASELESS (Sanchez-Arcilla et al., 2021) deals with different aspects of coastal downscaling, such as the importance of downscaling for including more physical processes and geometric features of the coastal areas (Trotta et al., 2021), the validation and error quantification of different data sources in the coastal environment (Cavaleri et al., 2018a; Wiese et al., 2018, Schulz-Stellenfleth and Staneva, 2019), and the improvement of the precision of numerical models either by coupling wind-wave-circulation models (Wiese et al., 2020) or by improving parameterizations of the physical processes (Du et al., 2019).
There are four sources of errors (or uncertainty) that ultimately introduce uncertainty in the model results when dynamically downscaling waves, namely (see Fig. 1): (a) forcing errors (or input errors), (b) model structural errors (or epistemic errors; i.e. due to simplifications or lack of knowledge in the model description of the real world), (c) model parameter errors (due to impossibility of defining a single set of parameter values), and (d) measurement errors. Forcing errors, particularly those related to boundary conditions, have a major impact on the downscaling results. Although global hindcasts have been significantly improved through better parametrizations of physical processes (see e.g. Filipot and Ardhuin, 2012 and Zieger et al., 2015) and through the use of better wind data from the most recent atmospheric reanalysis (e.g. Saha et al., 2014), errors are still expected, particularly related with swell far fields and high-order spectrum moments (Stopa et al., 2016). Event thought there are some correction methodologies that could alleviate the problem of input errors (e.g. Tomas et al. 2008; Mínguez et al., 2011, Albuquerque et al., 2018), they are mainly focused in correcting some aspect of the errors (e.g. total energy in the sea state) and there would be some errors remaining after the correction that deserve to be considered as a source of uncertainty when applying the model. Model structural and parameter errors are interrelated. Third-generation models based on the wave action balance equation are the state of the art for dynamically downscaling waves (e.g. SWAN, Booij et al., 1999), as the use of more accurate phase-resolving models is unfeasible given the time and spatial scales normally involved in the downscaling. However, for many of the physical processes involved in the sources and sink terms of the wave action balance equation, there are several possible parametrizations, the combination of which results in different model structures. Once a model structure has been selected, there is a set of parameters that must be defined. Even though there is prior knowledge on the range of values these parameters may take, and in some cases default values are proposed in the numerical models, many of the parameters would be case-specific and must be defined through calibration. Lastly, measurement errors arise from limitations of the measurement instruments, namely: accuracy, resolution and sampling frequency.
Assuming there is a set of observations available at the project site, the common practice is to fine-tune some of the model parameters in such a way that model outputs approximate as closely and consistently as possible the observed response of the system over the measured period. This approach assumes that model parameters are deterministic and neglects model structural errors as well as forcing and measurement errors. To the best of our knowledge, this calibration is usually addressed with ad hoc approaches developed for each project. As third-generation wave models have several parameters, the manual calibration of them becomes labor-intensive and strongly dependent on the modeler.
More advanced approaches resort to automatic calibration methods and would consider: (i) model parameters to be uncertain and to be handled as random variables and (ii) a parametric model for the correction of the boundary conditions, whose parameters are taken as part of the original model. These approaches allow to calibrate the model, thus improving its performance, and at the same time give an estimation of the uncertainty associated with its use. A variety of tools pursuing this has been developed and successfully applied in other branches of civil engineering, particularly in hydrology, using both formal (e.g. Vrugt et al., 2003, 2008 and 2016) and informal (e.g. Beven and Binley, 1992; 2006 and 2014) statistical approaches (see Section 2.1 for a review on this regard). Meanwhile, in coastal engineering, its application has been little exploited (Ruessink, 2005 and 2006, Alonso and Solari 2017, Simmons et al., 2017 and 2019, and Kroon et al., 2020; see section 2.2 for a review of previous works). In this work a methodology is proposed and tested for the automatic calibration of the free parameters of a wave propagation model and for the quantification of the uncertainty of the model result, when used for wave downscaling, with focus on uncertainties coming from input and parameters (sources (a) and (c) in Fig. 1).
The rest of the manuscript is organized as follows. First, section 2 reviews the state of the art in automatic calibration of numerical models free parameters and quantification of the uncertainty steaming from it in the framework of coastal engineering. Then, the objectives and methodology are presented in sections 3 Objectives, 4 Methodology, respectively, and the case study used for its application is introduced in section 5. Section 6 presents and discuss the obtained results. Finally, conclusions are outlined in section 7.
Section snippets
Background
This section reviews the state of the art of the statistical methods used to calibrate numerical models and to estimates its uncertainties, with focus on the methodologies previously applied in coastal engineering and on this work (Section 2.1), as well as the state of the art on their application to coastal engineering problems (Section 2.2).
Objectives
Starting from the early work of Alonso and Solari (2017), an improved methodology is proposed pursuing the following objectives:
- 1.
That the calibration improves model performance not only in terms of wave heights, but in terms of direction and periods as well;
- 2.
That it considers a parametric correction of the boundary conditions;
- 3.
That it runs fast enough for practical applications to be feasible.
Methodology
The proposed methodology includes five steps. First, the selection of a subset of data to calibrate the model, seeking to reduce the computational demand without losing representativeness of the variety of conditions present in the measured dataset. Second, the definition of a measure of how well the model fits the observations, establishing its statistical properties from which to define the likelihood function required for a formal approach. Third, the selection of the parameters to be
Study zone
The proposed methodology was applied to downscale the ERA-Interim (Dee et al., 2011) reanalysis in the Uruguayan Atlantic coast, where a few month wave measure data series is available (Fig. 3). Wave measurements were collected with an acoustic Doppler current profiler (ADCP) installed on a water depth of 18 m; measurements are 3-hourly for the period of October 2013–April 2014. The ERA-Interim node closest to the site is located about 100 km offshore, on a water depth of 60 m (53oW-35oS);
Results and discussion
Nine variables were used to define a sea state when applying the MDA, namely: significant wave height (Hs), peak period (Tp) and peak direction (Dp), for wind seas, southern swells and eastern swells. From the 665 sea states measured, 50 were selected with the MDA. Fig. 7 compares box plots obtained with the whole set of measured sea states with those obtained from the 50 sea states selected by the MDA. The similarity between the box plots demonstrate the ability of the MDA to provide a
Conclusions
A comprehensive methodology, based in the DREAM(zs) algorithm, was proposed and applied for automatic calibration and uncertainty estimation of a wave model when it is used in the framework of a dynamical downscaling of off-shore waves to a nearshore project site. Obtained results showed the ability of the methodology to reach a best-fit set of parameters for the wave model in an automatized way, and to provide uncertainty bands for downscaled wave parameters that could be useful when
Author statement
Rodrigo Alonso: Conceptualization, Methodology, Validation, Formal analysis, Data curation, Writing – original draft, Sebastián Solari: Conceptualization, Writing – review & editing, Supervision, Software.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The European Center for Medium-Range Weather Forecasts (ECMWF) is acknowledge for providing ERA-Interim surface wind and wave data. Uruguayan Ministry of Transport and Public Works (DNH-MTOP) is acknowledge for founding the wave measurement campaign, while Francisco Pedocchi and Rodrigo Mosquera are acknowledge for carrying out the campaign. This work was supported by the National Agency of Research and Innovation (ANII) in the frame of Project SUSME (ERANet-LAC 2nd Joint Call, project
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