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  • View in gallery

    Bathymetric map of the Yangtze River estuary, Hangzhou Bay, and the western part of the East China Sea, and the location of survey stations, including the in situ observational stations (red stars), the tide gauge station (red triangles), the stations for water level (red square) and the stations for flow (red dots). R1 is the area where the hydrodynamic model is run, and R2 is the area where the cohesive sediment transport model is run.

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    Comparison of flow velocity and direction between the modeled results (black solid line) and the observed results (red dots) at (a) S1 and (b) S2, and comparison of water level at (c) S3 and (d) S4.

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    (a) Initial SSCs at the surface layer and (b) mean values of the observations at every vertical layer (circles) and the fitted curve (solid line) vs the vertical number of the layer (1 for the bottom layer and 30 for the surface layer).

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    Variations of L1 norm of gradients of the cost function with respect to (a) settling velocity, (b) resuspension rate, and (c) initial condition in E02.

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    The AOs (red circles), COs (blue triangles), and the simulated results in E01 (black dashed line) and E02 (black solid line) at (a) S1, (b) S2, (c) S4, and (d) S7. Each panel contains three parts: (top to bottom) the surface, middle, and bottom parts of each station, respectively.

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    Spatially averaged temporal and vertical variabilities of (a) the current speeds and (b) the simulated SSCs in E02.

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    Spatially averaged temporal variability of the estimated settling velocity, which is smoothed with a 1-h moving filter in (a) E02 (bold black line), E11 divided by 1.5 (blue line), and E12 divided by 0.5 (red line); (b) E02 (bold black line), E21 (blue line), and E22 (red line); (c) E02 (bold black line), E31 (blue line), and E32 (red line); and (d) E02 (bold black line), E41 (blue line), and E42 (red line). Correlation coefficient R between the estimated settling velocity in E02 and that in the corresponding sensitivity experiment is shown.

  • View in gallery

    Spatially averaged temporal variability of the estimated resuspension rate, which is smoothed with a 1-h moving filter in (a) E02 (bold black line), E11 divided by 1.5 (blue line), and E12 divided by 0.5 (red line); (b) E02 (bold black line), E21 (blue line), and E22 (red line); (c) E02 (bold black line), E31 (blue line), and E32 (red line); and (d) E02 (bold black line), E41 (blue line), and E42 (red line). Correlation coefficient R between the estimated resuspension rate in E02 and that in the corresponding sensitivity experiment is shown.

  • View in gallery

    Spatially averaged temporal variability of (a) height above seabed (red line) and estimated settling velocity (blue line), (b) current speed (red line) and estimated settling velocity (blue line), (c) height above seabed (red line) and estimated resuspension rate (blue line), and (d) current speed (red line) and estimated resuspension rate (blue line) in E02, which are smoothed with a 1-h moving filter.

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A Methodology for Estimating the Parameters in Three-Dimensional Cohesive Sediment Transport Models by Assimilating In Situ Observations with the Adjoint Method

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  • 1 Physical Oceanography Laboratory/CIMST, Ocean University of China, and Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
  • 2 Institute of Physical Oceanography, Ocean College, Zhejiang University, Zhoushan, China
  • 3 State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Hangzhou, China
  • 4 School of Geographic and Oceanographic Sciences, Nanjing University, Nanjing, China
  • 5 State Key Laboratory of Marine Geology, Tongji University, Shanghai, China
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Abstract

The model parameters in the suspended cohesive sediment transport model are quite important for the accurate simulation of suspended sediment concentrations (SSCs). Based on a three-dimensional cohesive sediment transport model and its adjoint model, the in situ observed SSCs at four stations are assimilated to simulate the SSCs and to estimate the parameters in Hangzhou Bay in China. Numerical experimental results show that the adjoint method can efficiently improve the simulation results, which can benefit the prediction of SSCs. The time series of the modeled SSCs present a clear semidiurnal variation, in which the maximal SSCs occur during the flood tide and near the high water level due to the large current speeds. Sensitivity experiments prove that the estimated results of the settling velocity and resuspension rate, especially the temporal variations, are robust to the model settings. The temporal variations of the estimated settling velocity are negatively correlated with the tidal elevation. The main reason is that the mean size of the suspended sediments can be reduced during the flood tide, which consequently decreases the settling velocity according to Stokes’s law, and it is opposite in the ebb tide. The temporal variations of the estimated resuspension rate and the current speeds have a significantly positive correlation, which accords with the dynamics of the resuspension rate. The temporal variations of the settling velocity and resuspension rate are reasonable from the viewpoint of physics, indicating the adjoint method can be an effective tool for estimating the parameters in the sediment transport models.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jicai Zhang, jicai_zhang@163.com

Abstract

The model parameters in the suspended cohesive sediment transport model are quite important for the accurate simulation of suspended sediment concentrations (SSCs). Based on a three-dimensional cohesive sediment transport model and its adjoint model, the in situ observed SSCs at four stations are assimilated to simulate the SSCs and to estimate the parameters in Hangzhou Bay in China. Numerical experimental results show that the adjoint method can efficiently improve the simulation results, which can benefit the prediction of SSCs. The time series of the modeled SSCs present a clear semidiurnal variation, in which the maximal SSCs occur during the flood tide and near the high water level due to the large current speeds. Sensitivity experiments prove that the estimated results of the settling velocity and resuspension rate, especially the temporal variations, are robust to the model settings. The temporal variations of the estimated settling velocity are negatively correlated with the tidal elevation. The main reason is that the mean size of the suspended sediments can be reduced during the flood tide, which consequently decreases the settling velocity according to Stokes’s law, and it is opposite in the ebb tide. The temporal variations of the estimated resuspension rate and the current speeds have a significantly positive correlation, which accords with the dynamics of the resuspension rate. The temporal variations of the settling velocity and resuspension rate are reasonable from the viewpoint of physics, indicating the adjoint method can be an effective tool for estimating the parameters in the sediment transport models.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jicai Zhang, jicai_zhang@163.com

1. Introduction

Coastal and estuarine waters are often characterized by high concentrations of suspended sediments derived from the discharge of rivers or seabed resuspension (Miller and Mckee 2004). High suspended sediment concentrations (SSCs) will affect the productivity of upper-layer phytoplankton (Cloern 1987); the benthic ecosystem (Miller and Cruise 1995); the transport of carbon, nutrients, and pollutants (Mayer et al. 1998; Ilyina et al. 2006); and ocean engineering (Zhang et al. 2015). Suspended sediment transport is one of the most significant processes that will affect SSCs in the coastal and estuarine waters. An improved understanding of the physical process of the suspended sediment transport is of great importance for accurately predicting the SSCs and the associated morphology change and biogeochemical cycle. In situ measuring is the most direct method for investigating the mechanism of suspended sediment transport; however, it provides only a local description (Amoudry and Souza 2011). The polar-orbiting satellite ocean color data (e.g., Myint and Walker 2002; Warrick et al. 2004; Petus et al. 2010; Zhang et al. 2010) and geostationary satellite data (e.g., Salama and Shen 2010; Neukermans et al. 2012; He et al. 2013) can be used to map SSCs in coastal regions, but only the surface SSCs can be obtained, without three-dimensional structure. Numerical suspended sediment transport models have gradually become a powerful and useful tool to investigate the distribution and dynamics of suspended sediment (e.g., Wang and Pinardi 2002; Guan et al. 2005; Leupi et al. 2008; Hu et al. 2009; Amoudry 2014).

Except for some theoretical studies, the ocean models are supposed to be a temporally and spatially varying system (Guy and Fogelson 2008). Specially, the parameters in the models can be influenced, or decided, by the conditions of the ocean system, which indicate that these parameters should also be space and time dependent, for example, the ice–ocean drag coefficient (Lu et al. 2011), the bottom friction coefficient (Ludwick 1975; Wang et al. 2004; Bricker et al. 2005; Lozovatsky et al. 2008), and wind drag coefficient (Powell et al. 2003; Hwang 2005; Jarosz et al. 2007). The suspended sediment transport models include various physical processes, such as diffusion, deposition, erosion, and resuspension, which introduce large spatial and temporal variations into the system. In addition, the parameters in the suspended sediment transport models are functions of the local environments (e.g., the hydrodynamic background field, the SSCs, the bed density, and the sizes of sediment particles), which generally vary in space and time domain. Therefore, the model parameters should be also highly spatially and temporally varying, which can be proved by the observations shown in previous works. Agrawal and Pottsmith (2000) analyzed the measurements of particle size and settling velocity off the New Jersey coast and found that the settling velocity ws was a function of the particle radius an as follows: ws = 0.45 × 10−3an1.2. As indicated by Xia et al. (2004), the settling velocity of flocs (ws) depended on the size d, related by the equation ws = k × dn, where k and n are empirically determined constants. Many observations have revealed the values of n vary temporally and spatially (e.g., Fennessy et al. 1994; Dyer et al. 1996; Xia et al. 2004). Manning and Schoellhamer (2013) studied the properties of flocculated cohesive sediment through the data from a 147-km longitudinal transect in San Francisco Bay in California. They found that the best predictor of settling velocity was the flow velocities 39 min prior to sampling. Wang et al. (2013) presented the in situ measurements of hydrodynamics, sediment resuspension, and sediment particle size distribution for the Jiulong River estuary in southeastern China, and found that the settling velocity of aggregates was proportional to turbulence dissipation on a log scale.

The parameters in the sediment transport models are commonly parameterized based on experience or results from laboratory experiments. In most previous works, these parameters, especially the settling velocity and the resuspension rate, are usually set to be constant, which might introduce large modeling error. As indicated by the previous paragraph, it would be much more reasonable to suppose the model parameters to be space and time dependent, which can better reproduce the physical processes and improve the numerical simulation results. Nevertheless, the spatial and temporal distributions of model parameters are not easy to prescribe. Data assimilation methods have been proved to be able to effectively minimize the model-data misfits (Anderson et al. 1996) and to estimate the spatial and temporal variations of model parameters (Navon 1998).

As a four-dimensional variational data assimilation (4DVAR) method, the adjoint method has been widely applied to estimate the parameters in atmospheric and oceanographic models (e.g., Lu and Zhang 2006; Elbern et al. 2007; Kurokawa et al. 2009; Chen et al. 2014; Gao et al. 2015). Navon (1998) presented a significant overview on the parameter estimation in meteorology and oceanography in view of applications of 4DVAR to inverse parameter estimation problems. The adjoint method has also been implemented to estimate the poorly known parameters in the cohesive sediment transport models with the twin experiments (e.g., Yang and Hamrick 2003; Wang et al. 2017). Theoretically, the optimization of model parameters, even when they are assumed to be spatially and temporally varying, can be achieved by assimilating the in situ observations, which activates the present work.

Wang et al. (2017) developed a 3D sigma-coordinate cohesive sediment transport model with the adjoint data assimilation and investigated the parameter sensitivity of the model. They found that the model was sensitive to the initial condition, the inflow open boundary condition, the settling velocity, and the resuspension rate but insensitive to horizontal and vertical diffusivity coefficients; in addition, the constant sensitive parameters can be estimated more easily than the insensitive parameters in ideal twin experiments. In this study, based on the adjoint cohesive sediment transport model developed in Wang et al. (2017), the model parameters, including the settling velocity, the resuspension rate, and the initial condition, are estimated synchronously by assimilating in situ observations of SSCs in practical experiments. The temporal variations of the estimated settling velocity and resuspension rate are analyzed. The details of the rest of the paper are as follows: the in situ observations and the models are described in section 2, the modeling results and discussion are given in section 3, and the conclusions can be found in section 4.

2. The observations and models

a. The in situ observations

The in situ SSCs were observed at four stations, including S1, S2, S4, and S7 (Fig. 1), north of Hangzhou Bay. The observed SSCs at S1 and S2 started at 1000 UTC 9 May and ended at 1012 UTC 10 May 2005. The observed SSCs at S4 and S7 were during the period 0500 UTC 8 May–0700 UTC 9 May 2005. During the observing time, the tidal conditions were spring tide. The water samples at these stations were collected about hourly from three to five layers (from the surface to the bottom). The samples were then used to derive SSCs through filtration by using 0.45-μm filters. A total of 321 effective SSC observations were obtained at all four observational stations.

Fig. 1.
Fig. 1.

Bathymetric map of the Yangtze River estuary, Hangzhou Bay, and the western part of the East China Sea, and the location of survey stations, including the in situ observational stations (red stars), the tide gauge station (red triangles), the stations for water level (red square) and the stations for flow (red dots). R1 is the area where the hydrodynamic model is run, and R2 is the area where the cohesive sediment transport model is run.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

From the observations it is shown that the occurrence of resuspension and the measured SSCs at four stations had the same periodicity and that the maximum concentration lagged behind the maximum resuspension flux, indicating that the resuspension process represented an important factor controlling the concentration within the water column.

b. The cohesive sediment transport model

The governing equations for the evolution of SSCs are described as follows (Wang et al. 2017):
e1
e2
e3
e4
e5
e6
e7
where C represents the SSC; t is the time; x, y are the horizontal coordinates; σ is the vertical coordinate (0 for the bottom and 1 for the surface); H is the total water depth, including the undisturbed water depth and the sea surface elevation; u, υ, and w are the flow velocity components in the x, y, and σ directions, respectively; KH and KV are the horizontal and vertical diffusivity coefficients, respectively; ws denotes the suspended cohesive sediment settling velocity; Ω1 is the inflow open boundary; Ω2 is the outflow open boundary; Ω3 is the solid boundary; Cobc denotes the SSC at the inflow open boundary; is the normal to the boundary; C0 is the initial condition; and E and D are the erosion rate and the deposition rate, respectively. The erosion rate and the deposition rate are described as (Yang and Hamrick 2003; Wang et al. 2017)
e8
e9
where M0 is the resuspension rate; τb is the bottom shear stress; τce and τcd are the critical shear stress for erosion and deposition, respectively; and C1 is the SSC near the bottom.

Equations (1)(9) describe the suspended cohesive sediment transport model (forward model), and the finite difference schemes are given in Wang et al. (2017).

c. The adjoint model

To construct the discrete adjoint model, the cost function is defined as (Wang et al. 2017)
e10
where and are the simulated and observed SSCs at the (i, j, k) gird point at the nth time step, respectively. Term is the weighting matrix and theoretically should be the inverse of the observation error covariance matrix, which can be simplified assuming that the errors of the data are uncorrelated and equally weighted (Yu and O’Brien 1992). In the present study, the elements in are 1 where observations are available and 0 otherwise.
Based on the theory of the Lagrangian multiplier method, the Lagrangian function is defined as
e11
where λ denotes the adjoint variable of C; K is the total number of vertical layers, and there is one false layer at the bottom, numbered 0, while K + 1 represents the false layer at the surface.
To minimize the cost function J, the first-order derivate of the Lagrangian function with respect to the variables and parameters should be zero (Thacker and Long 1988),
e12
e13
e14
where p denotes the parameters in the cohesive sediment transport model, including KH, KV, ws, M0, Cobc, τce, τcd, and C0. In fact, Eq. (12) is the same as the discretization of Eq. (1). From Eq. (13), the discrete adjoint model can be derived. From Eq. (14), the gradients of the cost function with respect to parameters are obtained. The detailed formulas of the adjoint model and the gradients are listed in Wang et al. (2017).

In this work, the minimization algorithm used in the adjustment of parameters is the steepest descent method, which has been proved to be effective for the present model (Wang et al. 2017). As indicated by Gunzburger (2000), there are several stop criteria to end the iterations in the process of parameter estimation, including the cost functions of the last two steps are sufficiently small and close, the gradient is sufficiently small, the difference between the old and new guess values for the parameter is sufficiently close, or a combination of these criteria. In this work, the stop criterion is that the difference of cost functions, normalized by its initial value, between the last two steps is less than 2.0 × 10−5, with a maximum value of 500 for the iteration steps.

d. Model settings

To drive the sediment transport model, the hydrodynamic background field was calculated by the ROMS (Song and Haidvogel 1994) in the Yangtze River estuary, Hangzhou Bay, and the western part of the East China Sea (area R1 in Fig. 1). The horizontal orthogonal grids were applied to cover the model domain, in which the horizontal resolution was 0.0156° × 0.0128°. There were 30 uniform σ layers in the vertical direction. The month-average runoffs of the Yangtze River and the Qiantang River were given at the upstream boundaries. The temperature and salinity were not calculated in the hydrodynamic model. At the open boundaries, water level variations induced by astronomical tides were adopted, and the tidal harmonic constants were obtained from Oregon State University Tidal Inversion Software (Egbert and Erofeeva 2002). Four dominating constituents in the study area were considered, including M2, S2, K1, and O1. The harmonic constants of each constituent, analyzed from the calculated water levels of 60 days, were compared with the observed data at the tide gauge stations, which are shown in Fig. 1. At all the stations, the relative errors for the amplitude and the phase lag were less than 10%. The simulated depth-averaged flow velocity and direction during the spring tide condition, which was the same as that during the observing time of SSCs, agreed well with the observations at stations 1 and 2 (Figs. 2a and 2b); in addition, the simulated water levels of two stations—3 and 4—were in good agreement with the observations, especially at the spring tide (Figs. 2c and 2d). The results indicated that the calculated hydrodynamic background field can be used to force the cohesive sediment transport model.

Fig. 2.
Fig. 2.

Comparison of flow velocity and direction between the modeled results (black solid line) and the observed results (red dots) at (a) S1 and (b) S2, and comparison of water level at (c) S3 and (d) S4.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

In this work, the study area of the cohesive sediment transport model was Hangzhou Bay (area R2 in Fig. 1). The spatial resolutions of the sediment transport model were the same as the hydrodynamic model. The forward model was launched at 0500 UTC 8 May 2005. The integral time step was 240 s, and the total simulation time was 53.2 h, which were the same as those in the adjoint model. The flow velocities (u, υ, w) and the total water depth (H) were extracted from the hydrodynamic background field to force the sediment transport model.

The initial conditions are critical for the cohesive sediment transport models. Cai et al. (2015) found that surface SSCs had a negative correlation with the bathymetry in Hangzhou Bay. According the approximated relationship between surface SSCs and the water depth in Cai et al. (2015), the preliminary surface initial conditions were calculated. The mean values of the observed SSCs at the vertical model layers were linearly decreased with the vertical number of the layers (1 for the bottom layer and 30 for the surface layer), as shown in Fig. 3b. By adjusting the preliminary surface SSCs with the mean values of surface SSCs obtained from the fitted curve in Fig. 3b, the final distribution for the initial surface SSCs were obtained and shown in Fig. 3a. According to the relationship between the mean values of observed SSCs and the vertical number of the layers, the initial conditions at the other layers were given by multiplying the surface initial conditions with coefficients (6.4328 – 0.1811 × k), where k was the vertical number of the layer, and the values of k were equal to 1 and 30 for the bottom layer and the surface layer, respectively.

Fig. 3.
Fig. 3.

(a) Initial SSCs at the surface layer and (b) mean values of the observations at every vertical layer (circles) and the fitted curve (solid line) vs the vertical number of the layer (1 for the bottom layer and 30 for the surface layer).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

The western open boundaries were taken as the inflow open boundary, at which the input SSCs were set to 1.5 kg m−3 according to Du et al. (2010). In addition, the eastern open boundaries were taken as the outflow open boundary. The default value of the settling velocity was set to 1.0 × 10−5 m s−1, which agreed with the range of the settling velocity mentioned in Yang and Hamrick (2003), while the value of the resuspension rate was set to 2.0 × 10−6 kg m−2 s−1, which was consistent with the range mentioned in Ge et al. (2015). The critical erosion stress and the critical deposition stress were equal in the present model, where both were set to 0.1 N m−2, consistent with the range of critical stress in Hu et al. (2009). According to the dimensional analysis, the horizontal and vertical diffusivity coefficients were set to 80.0 and 1.5 × 10−4 m2 s−1, respectively.

3. Modeling results and discussion

As shown above, there are many model parameters poorly known in the present sediment transport model, and these parameters can vary over a wide range. As indicated by the previous works, the settling velocity and the resuspension rate can vary over the range of 10−7–10−3 m s−1 (Yang and Hamrick 2003) and 10−6–10−3 kg m−2 s−1 (Ge et al. 2015), respectively, depending on the different characteristics of the suspended sediment and hydrodynamic conditions. Yang and Hamrick (2003) pointed out that the settling velocity and the resuspension rate were two critical parameters controlling the sediment exchange process between the water column and the sediment bed, which was also shown in the results of the parameter sensitivity analysis in Wang et al. (2017). In addition, the initial values of SSCs are important for the simulation over short periods. Therefore, the settling velocity, the resuspension rate, and the initial condition will be estimated synchronously by assimilating the in situ observed SSCs in the following numerical experiments.

a. Data assimilation experiment

As concluded by Elbern et al. (2007), Zhang and Chen (2013), and some other works, the validity of the data assimilation should be tested by independent observations. In this study, some observations were randomly selected as “checking observations” (COs), which were not assimilated but used only to verify the assimilation results. The other observations were taken as the “assimilating observations” (AOs), which were assimilated into the model.

To verify the effect of data assimilation, a benchmark experiment with index E01 was carried out. In E01, the data assimilation was not used, and all the model parameters were set to the default values. In E02, the data assimilation was applied, and the number of COs was 16, which was about 5% of the total number of observations. The initial guess values of the parameters were set to the default values, which were the same as those in E01. The detailed model settings are shown in Table 1.

Table 1.

Detailed model settings of the experiments.

Table 1.

The L1 norm of gradients of the cost function with respect to the estimated parameters, including the settling velocity, the resuspension rate, and the initial condition, were reduced stably and largely during the iteration steps (Fig. 4), indicating that the adjustments of the model parameters were reasonable and effective. According to the vertical locations, the observations and the simulated results in E01 and E02 were divided into three parts, including the bottom (layers 1–10), the middle (layers 11–20), and the surface (layers 21–30) parts. The mean values of every part at the four in situ observational stations are shown in Fig. 5. Apparently, the simulated results in E02 were much closer to the observations and reproduced the temporal variations of the observed SSCs, which were better than those in E01. Both the observed SSCs and the simulated SSCs in E02 were larger for the lower layers. The mean absolute errors (MAEs) between the simulated SSCs and the AOs, and the MAEs between the simulated SSCs and the COs in E01 and E02 are presented in Table 2. The MAE between the simulated SSCs and the AOs over the four stations was 0.342 kg m−3 in E01 and 0.182 kg m−3 after data assimilation in E02, indicating that the model performance improved with a reduction of 46.78% in overall simulation error when the data assimilation was used. Besides, the MAE between the simulated SSCs and the COs at all four stations was 0.351 kg m−3 in E01 and 0.181 kg m−3 after data assimilation in E02, further showing a significant improvement when the data assimilation was used.

Fig. 4.
Fig. 4.

Variations of L1 norm of gradients of the cost function with respect to (a) settling velocity, (b) resuspension rate, and (c) initial condition in E02.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

Fig. 5.
Fig. 5.

The AOs (red circles), COs (blue triangles), and the simulated results in E01 (black dashed line) and E02 (black solid line) at (a) S1, (b) S2, (c) S4, and (d) S7. Each panel contains three parts: (top to bottom) the surface, middle, and bottom parts of each station, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

Table 2.

Statistics of the experiments before and after data assimilation.

Table 2.

The current speeds and simulated SSCs in E02 were horizontally averaged at every sigma layer and every time step; meanwhile, the total water depth was horizontally averaged. Then, the averaged current speeds and SSCs were interpolated from the sigma coordinate to the z coordinate using the mean water depth. So, the spatially averaged temporal and vertical variabilities of current speeds and SSCs were obtained, which is shown in Fig. 6. A phase difference between the current speed and the water level was produced, where minimum flow occurred after the high or low water level. The current speeds were much higher near the sea surface, with maximum values larger than 1.10 m s−1. The speeds were reduced close to the seabed due to friction, with minimum values of about 0.28 m s−1. The vertical SSCs indicated high concentrations appeared near the sea bed, being about double of the SSCs near the sea surface. The times series of the SSCs showed that high concentrations occurred during the flood tide and near the high water, which was similar to the observations in Wang et al. (2013). Wang et al. (2013) attributed this to the higher current speeds occurring at the tidal cycle. Comparing the current speeds with those in Wang et al. (2013), we found that the current speeds in this study were approximately equal to the highest current speed in Wang et al. (2013). In addition, the time series of the SSCs presented a clear semidiurnal variation. He et al. (2013) measured the water turbidity using a buoy near Hangzhou Bay and found the same phenomenon. It is further indicated that the variability of the simulated hydrodynamic background field and the simulated SSCs in E02 were in good agreement with the patterns obtained in other literatures, showing that the modeling results were reasonable.

Fig. 6.
Fig. 6.

Spatially averaged temporal and vertical variabilities of (a) the current speeds and (b) the simulated SSCs in E02.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

b. Sensitivity experiments

It is noted that the magnitude of smoothness (i.e., the step size in the steepest descent method), the initial guess values, and the assimilated observations may affect the results of the data assimilation and parameter estimation. In this section, several numerical experiments were carried out to study the sensitivity of the modeling results to these factors.

In E11 and E12, the initial guess values of the parameters were investigated. In the former experiment, the initial guess values of the settling velocity and the resuspension rate were increased by 50%, while they were decreased by 50% in the latter one. In E21 and E22, the number of COs was still 16, but the COs were randomly selected again. In E31 and E32, there were 32 observations (about 10% of the total observations) that were randomly selected as COs. In E41 and E42, the step sizes in the steepest descent method were increased by 50% and decreased by 50%, respectively. For the aforementioned experiments, the other model settings were the same as those in E02. The detailed model settings are listed in Table 1.

The cost functions normalized by the initial values at the first iteration step, the MAEs between AOs and the simulated results, and the MAEs between COs and the simulated results in these sensitivity experiments are shown in Table 2. All the normalized cost functions were less than 0.4, indicating that the observations had been assimilated into the model successfully. In addition, the MAEs between AOs and the simulated SSCs were decreased by at least 43.27%, while the MAEs between COs and the simulated SSCs were decreased by 15.77%, showing that the data assimilation improved the accuracy of the simulation of SSCs in all the experiments. The spatially averaged temporal variabilities of the estimated settling velocity and the resuspension rate, which were smoothed with a 1-h moving filter in the sensitivity experiments, are illustrated in Figs. 7 and 8, respectively. Although the values of the estimated settling velocity were not completely equal with that in E02, the variation tendencies were the same, which was the same as the estimated resuspension rate. In addition, all the correlation coefficients between the estimated parameters in E02 and those in the sensitivity experiments were larger than 0.8, indicating that the relationships between the estimated model parameters in E02 and those in the sensitivity experiments were significantly correlated. Therefore, the temporal variation tendencies of the estimated settling velocity are approximately the same for all of the experiments and were not affected by the model settings, which are the same as for the estimated resuspension rate. In other words, the estimated settling velocity and the estimated resuspension rate in E02 are robust and can reveal the temporal variations of the corresponding parameters in the present model.

Fig. 7.
Fig. 7.

Spatially averaged temporal variability of the estimated settling velocity, which is smoothed with a 1-h moving filter in (a) E02 (bold black line), E11 divided by 1.5 (blue line), and E12 divided by 0.5 (red line); (b) E02 (bold black line), E21 (blue line), and E22 (red line); (c) E02 (bold black line), E31 (blue line), and E32 (red line); and (d) E02 (bold black line), E41 (blue line), and E42 (red line). Correlation coefficient R between the estimated settling velocity in E02 and that in the corresponding sensitivity experiment is shown.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

Fig. 8.
Fig. 8.

Spatially averaged temporal variability of the estimated resuspension rate, which is smoothed with a 1-h moving filter in (a) E02 (bold black line), E11 divided by 1.5 (blue line), and E12 divided by 0.5 (red line); (b) E02 (bold black line), E21 (blue line), and E22 (red line); (c) E02 (bold black line), E31 (blue line), and E32 (red line); and (d) E02 (bold black line), E41 (blue line), and E42 (red line). Correlation coefficient R between the estimated resuspension rate in E02 and that in the corresponding sensitivity experiment is shown.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

c. Discussion: The temporal variations of the estimated settling velocity and resuspension rate

In this work, after data assimilation the modeling results of the SSCs have been improved. As mentioned above, the temporal variations of the settling velocity and resuspension rate are the main concern of this work. The sensitivity experiments indicate that the estimated parameters in E02 were not affected by the model settings. Therefore, in this section the estimated settling velocity and resuspension rate obtained from E02 are analyzed and discussed. The spatially averaged temporal variability of the height above the seabed, the current speed, and the estimated parameters (settling velocity and resuspension rate), smoothed with a 1-h moving filter, are shown in Fig. 9.

Fig. 9.
Fig. 9.

Spatially averaged temporal variability of (a) height above seabed (red line) and estimated settling velocity (blue line), (b) current speed (red line) and estimated settling velocity (blue line), (c) height above seabed (red line) and estimated resuspension rate (blue line), and (d) current speed (red line) and estimated resuspension rate (blue line) in E02, which are smoothed with a 1-h moving filter.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-16-0194.1

Figures 9a and 9b indicate that the estimated settling velocity correlated more strongly with the height above the seabed than that of the current speed. The correlation coefficient between the estimated settling velocity and the tidal elevation is −0.39 (N = 785), which is significant at the 99.9% confidence interval, inferring that the temporal variation of the estimated settling velocity is negatively correlated with the tidal elevation. In most cases, the maximal values of the estimated settling velocity occurred near the low water and the minimal values occurred about 1 h after the slack waters next to high waters.

A number of earlier research studies attempted to use Stokes’s law to relate the settling velocity of aggregates with the diameter of the aggregates and SSCs (e.g., Mikkelsen and Pejrup 2001; Winterwerp 2002). Besides, Voulgaris and Meyers (2004) used empirical formulas to establish the relationship between settling velocities and mean size of aggregates. In these studies, the settling velocity will be increased with increasing floc size. Wang and Xue (1990) pointed out that the suspended sediments in Hangzhou Bay were predominantly fine silt with less than 10% clay content; in addition, the sediment was finer in the northern region of the bay than the southern region, which was because the flood-dominated northern region was diluted by the intrusion of fine sediment from the Yangtze River estuary, whereas the southern region was influenced by the coarser material carried in from the Qiantang River. Therefore, for the whole bay the mean size of the suspended sediments becomes smaller during the flood tide, which decreases the settling velocity. This process will continue until the slack tide before ebb. During the time of ebb tide, the fine sediments will be carried out of the bay and the coarse suspended sediments from the Qiantang River will dominate the bay. At this condition, the mean size of the suspended sediments becomes coarser, which lead to the increasing settling velocity. The abovementioned analysis indicates that the estimated settling velocity using the data assimilation model is reasonable from the viewpoint of physics.

In Fig. 9d, the estimated resuspension rate became larger when the current speed was increased, while it became smaller for the decreased current speed. In addition, the maximal values of the estimated resuspension rate occurred when the current speed reached the maximum, and the minimal values occurred about 1 h after the minimal current speed. The relationship between the estimated resuspension rate and the height above the seabed was not obvious in Fig. 9c. The correlation coefficient between the estimated resuspension rate and the current speed is 0.44 (N = 785), which is significant at the 99.9% confidence interval.

As indicated by Ge et al. (2015), the resuspension rate was parameterized as a removal capability of soil material from the bed surface due to the current-induced shear stress. The large tidal currents can induce strong bottom shear stress and enhance sediment resuspension (He et al. 2013). Therefore, the resuspension rate and the current speed may have a positive correlation, indicating that the temporal variations of the estimated resuspension rate in E02 are also reasonable. The response of minimal estimated resuspension rate to the minimal current speed is delayed, possibly due to the effect of the slack water.

As mentioned in the introduction, the model parameters in the suspended sediment transport models can vary largely in space and time domain. In this work, the adjoint method has been proved to be a useful and effective approach that can estimate the model parameters in the suspended sediment transport models. There have been few works focused on the development of sediment transport models and parameter estimation with data assimilation technology. By assimilating the in situ observations, the adjoint method can improve the understanding of the temporal variations of the model parameters, which are important for the simulation of SSCs in coastal and estuarine waters.

4. Conclusions

Based on a three-dimensional sigma-coordinate suspended cohesive sediment transport model with the adjoint model, the in situ observed SSCs were assimilated to improve the results of the numerical simulation by estimating the model parameters. The results of the data assimilation experiments indicated that the data assimilation can significantly improve the numerical simulation results of SSCs, especially during the observed period. The time series of the modeled SSCs presented a clear semidiurnal variation, in which the high concentrations occurred during the flood tide and near the high water due to the high current speeds in Hangzhou Bay.

Through the sensitivity experiments, it was indicated that the estimated settling velocity and resuspension rate were not affected by the model settings, and they can reveal the temporal variations of the parameters in the present model. The temporal variation of the estimated settling velocity was negatively correlated with the tidal elevation due to the variations of the mean size of the suspended sediments, which was consistent with the results from Stokes’s law and the empirical formulas in Voulgaris and Meyers (2004). In addition, the maximal values of the estimated settling velocity occurred near the low waters, while the minimal values occurred after the slack waters next to high waters. The temporal variations of the estimated resuspension rate and the current speeds had a significantly positive correlation, which accorded with the physical meaning of the resuspension rate. The maximal values of the estimated resuspension rate occurred when the maximal current speeds occurred, while the minimal values occurred after the minimal current speeds. To sum up, the adjoint data assimilation can improve the results of the numerical simulation results of SSCs. The estimated results of the settling velocity and the resuspension rate are reasonable from the viewpoint of physics, indicating the adjoint data assimilation can improve the understanding of the physical processes in the sediment transport models.

It is necessary to point out that the number of assimilated observations in the present study is restricted. Therefore, only the temporal variations of the estimated settling velocity and the estimated resuspension rate are analyzed. In the future, we will try to assimilate the SSCs derived from remote sensing data, like the Geostationary Ocean Color Imager (Ryu et al. 2011), to further improve the results of data assimilation and parameter estimation, which will further improve the understanding of the physical processes of the sedimentation–resuspension cycle and the predication of SSCs in coastal and estuarine waters.

Acknowledgments

The authors thank the two anonymous reviewers for the constructive comments on the earlier version of the manuscript. This work is jointly supported by the Natural Science Foundation of Zhejiang Province through Grant LY15D060001; the National Key Research and Development Plan of China through Grants 2017YFC1404000, 2017YFA0604100, and 2016YFC1402304; the Key Research and Development Plan of Shandong Province through Grant 2016ZDJS09A02; and the National Natural Science Foundation of China through Grant 41206001.

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