Coastal hydrodynamics can be greatly affected by the presence of submerged aquatic vegetation. The effect of vegetation has been incorporated into the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system. The vegetation implementation includes the plant-induced three-dimensional drag, in-canopy wave-induced streaming, and the production of turbulent kinetic energy by the presence of vegetation. In this study, we evaluate the sensitivity of the flow and wave dynamics to vegetation parameters using Sobol' indices and a least squares polynomial approach referred to as the Effective Quadratures method. This method reduces the number of simulations needed for evaluating Sobol' indices and provides a robust, practical, and efficient approach for the parameter sensitivity analysis. The evaluation of Sobol' indices shows that kinetic energy, turbulent kinetic energy, and water level changes are affected by plant stem density, height, and, to a lesser degree, diameter. Wave dissipation is mostly dependent on the variation in plant stem density. Performing sensitivity analyses for the vegetation module in COAWST provides guidance to optimize efforts and reduce exploration of parameter space for future observational and modeling work.
The presence of aquatic vegetation (e.g., mangroves, salt marshes, and seagrass meadows) provides several ecological benefits including nutrient cycling, habitat provision, and sediment stabilization (Costanza et al., 1997). Vegetation provides a habitat for many species of epiphytes, invertebrates, and larval and adult fish (Heck et al., 2003). Seagrass meadows reduce sediment resuspension, thereby stabilizing bottom sediment, increasing light penetration, and improving water clarity in a positive feedback loop (Carr et al., 2010). In addition, aquatic vegetation provides coastal protection by absorbing wave energy (Wamsley et al., 2010).
One approach to implement the influence of aquatic vegetation on circulation is by increasing the bottom roughness coefficient (Ree, 1949; Morin et al., 2000). Recent studies involving 2-D depth-averaged models (Chen et al., 2007; Le Bouteiller and Venditti, 2015) have quantified the effect of vegetation through parameterization as “form drag” as opposed to “skin friction”. To account for 3-D vertical structures, estuary-scale models have implemented both mean and turbulent flow impacts of vegetation (Temmerman et al., 2005; Kombiadou et al., 2014; Lapetina and Sheng, 2014). In addition to impacts on the flow field, the presence of vegetation also results in wave attenuation. The decay of wave height over vegetation has been simulated by enhancing bed roughness (Möller et al., 1999; de Vriend, 2006; Chen et al., 2007). A more physical description of wave attenuation due to vegetation was developed by Dalrymple et. al (1984), who approximated wave energy loss due to stalks approximated as cylinders. This approach has been applied in spectral wave models and calibrated against flume experiment results (Mendez and Losada, 2004; Suzuki et al., 2012; Wu, 2014; Bacchi et al., 2014).
Recently, Beudin et al. (2017) implemented the effects of vegetation in a vertically varying water column through momentum extraction and turbulence dissipation and generation using a 3-D hydrodynamic model and accounting for wave dissipation due to vegetation in a spectral wave model. The modeling approach was implemented and tested within the open-source COAWST (Coupled Ocean–Atmosphere–Wave–Sediment Transport) modeling system that couples hydrodynamic and wave models (Warner et al., 2010). The vegetation module was based on modifications to the flow field resulting from three-dimensional drag, in-canopy wave-induced streaming, and production of turbulent kinetic energy in the hydrodynamics model (Regional Ocean Modeling System – ROMS), along with energy dissipation and resultant hydrodynamic feedback from the wave model (Simulating WAves Nearshore – SWAN).
The vegetation module requires the user to input a given set of plant properties (stem density, height, diameter, and thickness). These vegetation properties can be highly variable depending on the season and environment, yet obtaining a full set of measurements in realistic settings is impractical. Identifying which properties have the greatest influence on the resulting flow dynamics can reduce the amount of observational data required to robustly parameterize the model and/or reduce the number of runs required in a model ensemble to quantify the uncertainty associated with data gaps. Our study aims to perform a systematic sensitivity analysis to quantify the effect of changing the vegetation properties on the resulting hydrodynamic output. The results of the sensitivity analysis can be used to select and rank the most important parameters for calibration. Two conditions are required for the model to display a significant sensitivity: (1) a sufficient modification of one of the forcing parameters and (2) a change in the leading terms of the dynamic equations of the model. While modifying the forcing parameters by a sufficient amount is required, the modification should remain within the natural range of variability of the parameters.
Several mathematical techniques have been utilized to perform sensitivity analysis. Bryan (1987) applied scaling analysis to an idealized domain and forcing, and found that closure parameters such as vertical diffusivity and wind stress curl were important controlling factors in thermohaline circulation. Bastidas et al. (1999) used multicriteria methods to find the sensitivity of land surface scheme models that couple biosphere–atmosphere interactions. The input variables (such as precipitation, air temperature and humidity, etc.) predict the evolution of soil skin temperature, soil moisture, etc. The input parameters obtained from the sensitivity analysis of the model showed consistency with physical properties for two different field sites and helped to identify insensitive parameters that led to an improvement in model description. Fennel et al. (2001) incorporated adjoint methods to perform sensitivity studies to refine ecological model parameters such that the underlining model can be applied to a wider range of conditions. Mourre et al. (2008) performed multiple simulations based on realistic variation of a forcing field to calculate the influence of model parameters on sea surface salinity. The metric used to measure the sensitivity was based on rms difference between the reference and modified model parameter. The results showed that lateral salt diffusivity had the strongest impact on surface salinity model response. Rosero et al. (2010) investigated the sensitivity of three different versions of a land satellite model (Noah LSM) applied to nine different sites based on different conditions (soil, vegetation, and climate). They utilized the Monte Carlo method to generate the first-order Sobol' indices (Sobol', 1993) to estimate the model sensitivity. The results showed that the optimal parameter values varied between different versions of the models and for different sites along with quantifying the nature of interactions between parameters. One of the challenges associated with a Monte Carlo approach to computing the Sobol' indices is the large number of model evaluations required for approximating conditional variance.
All these studies highlight various approaches to perform sensitivity analysis. Saltelli et al. (2008) provided a comparison of different sensitivity analysis methodologies and the optimal setup for specific combinations of parameters and model. Ultimately, the choice of sensitivity analysis methodology depends on multiple factors such as the computational cost of running the model, the characteristics of the model (e.g., nonlinearity), the number of input parameters, and/or the potential interactions between parameters. Saltelli et al. (2008) described variance-based techniques as providing the most complete and general pattern of sensitivity for models with a limited number of parameters, such as the vegetation module in COAWST. Sobol' indices, as a form of variance-based sensitivity analysis, provide a decomposition of the variance of a model into fractions that can be assigned to inputs or combinations of inputs. However, techniques involving the estimation of Sobol' indices (such as Monte Carlo methods) are expensive.
Schematic showing the vegetation module implementation in the COAWST model (figure adapted from Beudin et. al, 2017).
To reduce the computational cost and have desirable accuracy, techniques that involve approximating the global response of the model with a polynomial and then using its coefficients to estimate the Sobol' indices can be utilized (Sudret, 2008). In this paper, we use a set of least squares polynomial tools based on subsampling to estimate our global polynomial response (Seshadri et al., 2017). Then, the coefficients of the polynomial are used to compute the Sobol' indices. These tools are implemented in the open-source package Effective Quadratures (EQ) method (Seshadri and Parks, 2017), and our current work provides one of the first applications of this methodology to quantify sensitivity of input parameters in coastal models. Therefore, the goal of the present work is to take advantage of the EQ method to provide Sobol' indices that quantify the sensitivity of the flow and wave dynamics to vegetation parameters in COAWST model. The paper is organized as follows: the methods are discussed in Sect. 2, including the numerical model with vegetation model (COAWST), the Effective Quadratures method to estimate Sobol' indices, and simulation design; in Sect. 3, we present the results of sensitivity analysis from various simulations; in Sect. 4, we discuss the impact of these results; and finally, in Sect. 5, we summarize our work and outline areas of future research.
Beudin et al. (2017) implemented a hydrodynamic-vegetation routine within the open-source COAWST numerical modeling system. The COAWST framework utilizes ROMS for hydrodynamics and SWAN for modeling waves coupled via the Model Coupling Toolkit (MCT) (Warner et al., 2008b).
ROMS is a three-dimensional, free surface, finite-difference, terrain-following model that solves the Reynolds-averaged Navier–Stokes equations using the hydrostatic and Boussinesq assumptions (Haidvogel et al., 2008). The transport of turbulent kinetic energy and generic length scale are computed with a generic length scale (GLS) two-equation turbulence model. SWAN is a third-generation spectral wave model based on the action balance equation (Booij et al., 1999). The effect of submerged aquatic vegetation in ROMS is to extract momentum, add wave-induced streaming, and generate turbulence dissipation. Similarly, the wave dissipation due to vegetation modifies the source term of the action balance equation in SWAN. Sub-grid-scale parameterizations account for changes due to vegetation in the water column extending from the bottom layer to the height of the vegetation in the flow model, while SWAN accounts for wave dissipation due to vegetation at the seafloor. The parameterization of SWAN to account for wave dissipation implemented by Suzuki et al. 2012 has the same effect as energy dissipation.
Processes in ROMS and SWAN to model the presence of
vegetation. The different input parameters (stem density,
The parameterizations used to implement the effect of vegetation in both ROMS and SWAN models are mentioned in Table 1 and detailed in Beudin et al. (2017). The coupling between the two models occurs with an exchange of water level and depth-averaged velocities from ROMS to SWAN and wave fields from SWAN to ROMS after a fixed number of time steps (Fig. 1). The vegetation properties are separately input in the two models at the beginning of the simulations.
Polynomial techniques are ubiquitous in the field of uncertainty quantification and model approximation. They estimate the response of some quantity of interest with respect to various input parameters using a global polynomial. From the coefficients of the polynomial, the mean, variance, skewness, and higher-order statistical moments can be calculated (see Smith, 2014; Geraci et al., 2016). In this paper, our interest lies in statistical sensitivity metrics called first-order Sobol' indices (Sobol', 1993) that are derived from the conditional variances of the parameters of the model. These indices are the same in number as the input parameters to the model and quantitatively rank the input parameters based on their contribution to the resultant model output. Thus, model output is more sensitive to parameters that exhibit higher first-order Sobol' index value. Second-order and third-order Sobol' indices may also be computed. The sum of the first-order, second-order, and third-order Sobol' indices should be equal to unity; therefore, if the first-order indices are themselves close to unity, it indicates the higher-order interaction between model input parameters is weak.
Schematic showing the idealized domain (not drawn to scale):
In this paper, the first-order indices are computed from a global polynomial model using the effectively subsampled quadratures method (ESQM v5.2; Seshadri and Parks, 2017). There are two attributes to any data-driven polynomial model: the choice of the polynomial basis and the strategy for estimating the coefficients of the polynomial. The basis used in Effective Quadratures is orthogonal polynomials, i.e., orthogonal with respect to the weight of the input parameter. For example, if one of the input parameters is prescribed with a Gaussian distribution, then a Hermite orthogonal polynomial basis would be used; likewise, for a uniform distribution, Legendre polynomials are used. In the current work, a uniform distribution is assumed for the input parameter space. The rationale behind selecting polynomials that are orthogonal with respect to the input weight is that it reduces the number of model evaluations required for estimating statistical moments. Details on the exponential convergence in moments when matching the orthogonal polynomial with its corresponding weight can be found in Xiu and Karniadakis (2002).
The coefficients for the polynomial expansion are typically approximated
using an integral over the input parameter space using quadrature rules.
When the number of input parameters is greater than one, tensor-grid or
sparse-grid-based quadrature rules may be used to approximate these
integrals. However, the cost of tensor grids grows exponentially with
dimension; i.e., a four-point quadrature rule in three dimensions has
5
The method of Effective Quadratures determines points for approximating the integral by subsampling well-chosen points from a tensor grid and evaluating the model at those subsamples. These well-chosen points are obtained via a QR column pivoting heuristic (Seshadri et al., 2017). Once the coefficients are estimated, the Sobol' indices can be readily computed (Sudret, 2008).
Prior to performing the simulations for estimating Sobol' indices described
above, a range of vegetation inputs that would impact the model response
needs to be chosen. Kennish et al. (2013) constrained annual variation of
three of the four vegetation properties (stem density, height, diameter)
based on Stem density Height Diameter Thickness
For the sensitivity analysis, a combination of these ranges of inputs (Table 2) is chosen to configure different simulations in an idealized test case
(described below). In addition to these four vegetation properties, the
vegetative model requires an input of drag coefficient
An idealized rectangular model domain of 10 km by 10 km with a 3 m deep
basin is chosen. The grid is 100 by 100 in the horizontal (100 m resolution)
and has 60 vertical sigma-layers (uniformly distributed) leading to 0.05 m resolution in the vertical. The vertical resolution of 0.05 m allows a plant
height of 0.27 m to be distributed over six vertical layers while the shortest
height is restricted to two vertical layers. A square patch of vegetation (1 km by 1 km) is placed in the middle of the domain (Fig. 2). The ROMS barotropic and
baroclinic time steps are, respectively, 0.05 and 1 s, while the SWAN time step and the
coupling interval between ROMS and SWAN are 10 min. The friction exerted on
the flow by the bed is calculated using the Sherwood–Signell–Warner bottom boundary layer (SSW-BBL) formulation (Warner et al., 2008a). The bottom boundary layer roughness
is increased by the presence of waves that produce enhanced drag on the mean
flow (Madsen, 1994; Ganju and Sherwood, 2010). The vegetative drag
coefficients
Plant property input combinations for different simulations during sensitivity analysis.
The model is forced by oscillating the water level on the northern edge with a tidal amplitude of 0.5 m and a period of 12 h. Waves are also imposed on the northern edge with a height of 0.5 m, directed to the south (zero angle), with a period of 2 s. The test case setup is similar to the one used by Beudin et al. (2017). The test case setup is simulated for 2 days to obtain a tidally steady state solution. These simulations require 40 CPU hours on Intel Xeon® X5650 2.67 GHz processors running on 24 parallel processors.
The output parameters used to investigate the vegetation model sensitivity
are chosen to reflect the first-order effects of vegetation on the
hydrodynamics and waves. The results for the model response are computed for
the last tidal cycle (a total of three tidal cycles are required for
achieving steady state). The presence of vegetation affects the output
parameters in different physical ways (Table 1).
Percentage change from minimum for the four impact parameters:
The response impact to change in the inputs during each simulation is computed by calculating the percentage difference of model response for each simulation from the minimum value of all the simulations. The model response is obtained in and around the vegetation patch and averaged over the last tidal cycle. The change in model response of water level is computed by finding the maximum water level difference in and around the vegetation patch. In addition, the variability of model response with given vegetation inputs in different simulations is calculated through standard deviation of model response in and around the vegetation patch over the last tidal cycle. The standard deviation in TKE is depth averaged to provide a 2-D field.
Using the range of input parameters described above (Sect. 2.3), and
assuming all the inputs are uniformly distributed over their ranges, a
matrix of design of experiment values (Table 2) was determined using
Effective Quadratures. A total of 15 simulations were found to be required,
corresponding to the number of coefficients in a 4-D polynomial with a
maximum order of 2. In general, the number of coefficients,
The 15 simulations (parameter choice of each simulation in Table 2) are
performed to provide model response from the four chosen output variables.
Standard deviation from wave dissipation (W m
Standard deviation in kinetic energy (cm
Standard deviation in water level in the presence of vegetation (plan view). The area of the vegetation patch is highlighted in the middle of the domain.
Following the variability in model response from different simulations, the
sensitivity to input vegetation parameters can be quantified with the use of
first-order Sobol' indices that are obtained by taking advantage of the
Effective Quadratures approach. Sobol' indices are individually computed for
all the model responses. The first-order Sobol' indices for all the model
responses (Table 3) add up to more than 0.9. This result indicates they
account for 90 % of the variability in model response for the given
vegetation property inputs, and the variability captured by second-order and
third-order indices is relatively low. The model is most sensitive (Table 3)
to plant stem density (
From the different simulations performed during sensitivity analysis, there
is a great amount of variability in front of the vegetation patch in wave
dissipation, kinetic energy (KE), and TKE (Figs. 4, 5, and 7). This is a result of large
amount of wave dissipation and flow deceleration in front of the vegetation
patch. The cross-sectional plane of the domain illustrates that the
variability in KE occurs throughout the water column (Fig. 5), highlighting
the 3-D impact of vegetation inputs. Interestingly, the greatest amount of
variability in KE (Fig. 5) occurs at distance above the bed between 0.4 and 1.3 m at
Standard deviation in TKE
(cm
Sobol' indices for all the outputs.
The parameterizations involving extraction of momentum, turbulence
production, and turbulence dissipation are directly affected by vegetation
stem density
The high sensitivity of wave dissipation to vegetation stem density highlights the need for accurate density representation to attain wave attenuation estimates, especially in open coasts. SWAN computes wave dissipation due to vegetation as a bottom layer effect. Therefore, the height of the vegetation does not affect wave dissipation to the same extent as other model outputs: KE, TKE, and water level. In addition, the equation representing the wave dissipation process in SWAN is independent of vegetation thickness, thus corresponding to the lowest Sobol' index. Vegetation thickness only appears in the turbulence dissipation term (Table 1); modifying the turbulence length scale has the least effect on any of the model responses.
To complement the results of the Sobol' indices' calculation, linear fits to
the data are conducted. The main parameter contributing to the wave
dissipation variability is the stem density, explaining over 80 % of the
variability (
The kinetic energy variability is also associated with stem density changes, but the percentage of explained variability (42 %) is smaller than for wave dissipation. Diameter and height also contributed to changes in kinetic energy. The combination of stem density, height, and diameter provided the optimal fit of the data (selected by minimizing BIC) and explained 89 % of variability in kinetic energy. Similar results were obtained for TKE, with density explaining 45 % of the variability but the combination of stem density, height, and diameter providing the optimal fit to TKE (selected by minimizing BIC) and explained 87 % of the variability. The model response of water level variability is also best explained by a combination of stem density, height, and diameter (96 % of water level variance explained). Thickness was not correlated with wave dissipation, kinetic energy, or TKE and only contributed to water level gradient variability.
The model configuration chosen includes vegetation covering a small fraction
of the water column to allow for proper wave dissipation. Many species of
seagrass have a larger vertical footprint and can also exhibit much higher
shoot densities. The goal of the study is to provide estimates on the
relative importance of the different parameters through a robust sensitivity
approach. The current work assumed rigid vegetation blades, while the model
is capable of including flexible vegetation by altering the blade scale. The
expected effect of flexible blades would be to reduce the relative
importance of vegetation length
The coupled wave-flow–vegetation module in the COAWST modeling system provides a tool to study vegetated flows in riverine, lacustrine, estuarine, and coastal environments. The resulting flow field in the presence of vegetation depends on its properties, including vegetation stem density, height, diameter, and thickness. The sensitivity of the hydrodynamic and wave conditions to changes in vegetation parameters is investigated. The sensitivity analysis helps in understanding the multi-parameter/multi-response of various interactions within the model. We use an existing tool that formulates the Effective Quadratures method to quantify the sensitivity of plant input properties for the vegetation module in COAWST model. The decomposition of the variance of the model solution given by the Sobol' indices is assigned to plant parameters.
The method of using Sobol' indices to quantify sensitivity can be computationally expensive. One of the goals of this work is to demonstrate a robust, practical, and efficient approach for the parameter sensitivity analysis. We show that the approach of using the Effective Quadratures method to select a parameter space that is consistent with physical understanding significantly reduces the computational time required to obtain the Sobol' indices.
The evaluation of Sobol' indices shows that the input values of plant stem density, height, and, to a lesser degree, diameter are consequential in determining kinetic energy, turbulent kinetic energy, and water level changes. Meanwhile, the wave dissipation is mostly dependent on the variation in plant density.
The sensitivity analysis for the vegetation model in COAWST presented herein provides guidance for observational and modeling work by allowing future efforts to focus on constraining the most influential inputs without having to explore the entire parameter space. An accurate representation of processes causing kinetic energy and turbulent kinetic energy leads to enhanced understanding of sediment processes while accurate water level computations help to predict coastal flooding caused by storm surge. Similarly, wave attenuation measurements in open coasts are better understood with a correct representation of wave dissipation. In the future, we intend to perform a similar sensitivity analysis with the inclusion of a biological model that will affect plant growth, thus allowing a time dependence of model input and response. In addition, the influence of vegetation on sediment transport will be explored. As model complexity increases with more parameters representing additional processes, input parameter sensitivity is required for the model to be applied in practical applications.
The Effective Quadratures methodology is an open-source
Python-based tool designed to perform sensitivity analysis for a given
physical system. The instructions to install the code along with all the
open-source files for this tool are detailed here:
The COAWST model is an open-source coupled hydrodynamics and wave model
containing vegetation effects mainly coded in Fortran 77. This model provided
the physical setting to perform the sensitivity analysis. The code is
available from
The model output from various simulations used to perform
sensitivity analysis in this study is available
at
TSK and NKG designed and simulated the numerical experiment space for sensitivity analysis. PS developed the Effective Quadratures methodology to quantify the model sensitivity. AB provided inputs on the mechanistic processes involving the vegetation model. TSK and AA performed the data analysis from the output of sensitivity study and prepared the paper with contributions from all co-authors.
The authors declare that they have no conflict of interest.
Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the US Government.
We thank Jeremy Testa at the University of Maryland Center for Environmental Science for providing us guidance on the ranges of vegetation for sensitivity studies in the early stages of work. We also thank P. Soupy Daylander at the US Geological Survey, St. Petersburg Coastal and Marine Science Center, for providing her feedback to improve the clarity of the paper. We thank the anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions. Edited by: Paul Halloran Reviewed by: two anonymous referees