Variational inverse parameter estimation in a cohesive sediment transport model: An adjoint approach

Suspended sediment dynamics play an important role in controlling nearshore and estuarine geomorphology and the associated ecological environments. Modeling the transport of suspended sediment is a complicated and challenging research topic. The goal of this study is to improve the accuracy of modeling the suspended sediment concentrations (SSCs) with newly developed techniques. Based on a three-dimensional suspended cohesive sediment transport model, the transport of suspended sediment and SSCs are simulated by assimilating SSCs retrieved from the Geostationary Ocean Color Imager (GOCI) with the adjoint data assimilation in the Hangzhou Bay, a typical strong tidal estuary along the coast of the East China Sea. To improve the effect of the data assimilation, the penalty function method, in which the reasonable constraints of the estimated model parameters are added to the cost function as penalty terms, will be introduced for the first time into the adjoint data assimilation in the SSCs modeling. In twin experiments, the prescribed spatially varying settling velocity is estimated by assimilating the synthetic SSC observations, and the results show that the penalty function method can further improve the effect of data assimilation and parameter estimation, regardless of synthetic SSC observations being contaminated by random artificial errors. In practical experiments, the spatially varying settling velocity is firstly estimated by assimilating the actual GOCI-retrieved SSCs. The results demonstrate that the simulated results can be improved by the adjoint data assimilation, and the penalty function method can additionally reduce the mean absolute error (MAE) between the independent check observations and the corresponding simulated SSCs from 1.44 × 10−1 kg/m3 to 1.30 × 10−1 kg/m3. To pursue greater simulation accuracy, the spatially varying settling velocity, resuspension rate, critical shear stress and initial conditions are simultaneously estimated by assimilating the actual GOCI-retrieved SSCs to simulate the SSCs in the Hangzhou Bay. When the adjoint data assimilation and the penalty function method are simultaneously used, the MAE between the independent check observations and the corresponding simulated SSCs is just 9.90 × 10−2 kg/m3, which is substantially less than that when only the settling velocity is estimated. The MAE is also considerably less than that when the four model parameters are estimated to be without using the penalty function method. This study indicates that the adjoint data assimilation can effectively improve the SSC simulation accuracy, and the penalty function method can limit the variation range of the estimated model parameters to further improve the effect of data assimilation and parameter estimation.

PAN, C. and HUANG, W., 2010. Numerical modeling of suspended sediment transport in Qiantang River: an estuary affected by tidal bore. Journal of Coastal Research, 26(6), 1123–1132. West Palm Beach (Florida), ISSN 0749-0208. Study of suspended sediment transport in an estuary affected by tidal bore is important for coastal engineering and management because the tidal bore can cause strong sediment resuspension and scour in shallow waters. Numerical modeling of suspended sediment transport in a natural estuary affected by tidal bore is such a challenging research topic that there are few articles on the subject available in the literature. In this study, a two-dimensional, numerical model was developed to investigate suspended sediment transport induced by a tidal bore. The hydrodynamic component of the model employs the Godunov-type scheme with second-order accuracy in space, which effectively describes the rapid supercritical flow and sharp horizontal pressure gradients of the tidal bore. To preserve balance between the source terms and the internal forces, both the water level–bottom topography formulation (WLTF) method and a special technique for triangular mesh have been applied to solve the source term in the model equations to account for the irregular bottom topography. The wet/dry boundary issue was solved by using the improved exact-Riemann solver on the dry bed. The coupled sediment transport model incorporates more reliable equations from recent publications to characterize the rapid increase of sediment resuspension in the water column. The model test against an analytical solution of convection transport shows that the sharp gradient of scalar transport is satisfactorily estimated in the model simulations. The model in the application has been validated to simulate hydrodynamics and suspended sediment transport affected by a tidal bore in the Qiantang River of China. The results compare well with a time series of observations to characterize the rapid increases of surface elevation, currents, and suspended sediment concentration resulting from the tidal bore. Results of spatial distributions of water levels and currents indicate that the model adequately describes the sharp horizontal gradients of the surface elevation and the tidal currents during the passage of the tidal bore and characterizes the suspended concentrations in the estuary.

Data assimilation is predominantly used for state estimation, combining observational data with model predictions to produce an updated model state that most accurately approximates the true system state whilst keeping the model parameters fixed. This updated model state is then used to initiate the next model forecast. Even with perfect initial data, inaccurate model parameters will lead to the growth of prediction errors. To generate reliable forecasts, we need good estimates of both the current system state and the model parameters. This article presents research into data assimilation methods for morphodynamic model state and parameter estimation. First, we focus on state estimation and describe implementation of a three‐dimensional variational (3D‐Var) data assimilation scheme in a simple 2D morphodynamic model of Morecambe Bay, UK. The assimilation of observations of bathymetry derived from synthetic aperture radar (SAR) satellite imagery and a ship‐borne survey is shown to significantly improve the predictive capability of the model over a 2‐year run. Here, the model parameters are set by manual calibration; this is laborious and is found to produce different parameter values depending on the type and coverage of the validation dataset. The second part of this article considers the problem of model parameter estimation in more detail. We explain how, by employing the technique of state augmentation, it is possible to use data assimilation to estimate uncertain model parameters concurrently with the model state. This approach removes inefficiencies associated with manual calibration and enables more effective use of observational data. We outline the development of a novel hybrid sequential 3D‐Var data assimilation algorithm for joint state‐parameter estimation and demonstrate its efficacy using an idealised 1D sediment transport model. The results of this study are extremely positive and suggest that there is great potential for the use of data assimilation‐based state‐parameter estimation in coastal morphodynamic modelling. Copyright © 2012 Royal Meteorological Society

Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

Sediments suspended during dredging operations impair water quality and can transport hydrophobic contaminants downstream. Suspended sediment transport models are available, but require an estimate of the rate of sediment resuspension resulting from the dredging operations. In this study, data from cutterhead suction dredging operations, an assessment of sediment resuspension mechanics during cutterhead suction dredging operations, and dimensional analysis are combined to develop empirical models of suspended sediment generation. The usefulness of the models for making a priori estimates of sediment resuspension rates associated with other cutterhead dredging operations is discussed. The sensitivity of the empirical models to variations in physical and operational parameters is investigated.

We propose a methodology for the treatment of the systematic model error in variational data assimilation. The principle of the method is to add a systematic error correction term in the model equations and to include it in the variational assimilation control vector.This method is applied to a simplified ocean circulation model in an identical twin experiment framework. It shows a noticeable improvement compared to the result of a classical variational assimilation scheme in which the systematic error is not corrected. The estimated systematic error correction term is sufficiently consistent with that needed by the model that it allows improvements not just to the analysis, but also during the forecast phase.

10.1016/0967-0653(95)96479-o

ABSTRACTFour-dimensional variational (4D-Var) data assimilation for operational systems requires the solution of large linear systems that are poorly conditioned in general. In addition to efficient iterative solvers for linear systems, using a good preconditioner is required to guarantee an acceptable solution with a small number of iterations. We consider the assimilation of ocean observations using a weak constraints 4D-Var for the Navy Coastal Ocean Model (NCOM), based on the representer method. Two methods of preconditioning the linear system are implemented, namely the scaling of the linear system by the square root of the observations error variances, and the approximating stabilised representer matrix based on the computation of some representer functions. We evaluate their convergence using criteria such as the norm of the residuals and of the gradient of the cost function, the analysis error evaluated at the observation locations, and finally the convergence of the sequence of analyses. Results from all criteria show that the fastest convergence is achieved with the rescaling preconditioner when the conjugate gradient used to solve the linear system is equipped with a suitable inner product instead of the Euclidean.

At the National Centers for Environmental Prediction, the global ensemble forecasts from the ensemble Kalman filter scheme in the Global Forecast System are applied in a regional three-dimensional (3D) and a four dimensional (4D) ensemble–variational (EnVar) data assimilation system. The application is a one-way variational method using hybrid static and ensemble error covariances. To enhance impact, three new features have been added to the existing EnVar system in the Gridpoint Statistical Interpolation (GSI). First, the constant coefficients that assign relative weight between the ensemble and static background error are now allowed to vary in the vertical. Second, a new formulation is introduced for the ensemble contribution to the analysis surface pressure. Finally, in order to make use of the information in the ensemble mean that is disregarded in the existing EnVar in GSI, the trajectory correction, a novel approach, is introduced. Relative to the application of a 3D variational data assimilation algorithm, a clear positive impact on 1–3-day forecasts is realized when applying 3DEnVar analyses in the North American Mesoscale Forecast System (NAM). The 3DEnVar DA system was operationally implemented in the NAM Data Assimilation System in August 2014. Application of a 4DEnVar algorithm is shown to further improve forecast accuracy relative to the 3DEnVar. The approach described in this paper effectively combines contributions from both the regional and the global forecast systems to produce the initial conditions for the regional NAM system.

A four-dimensional variational data assimilation (4DVar) system of the LASG/IAP Climate Ocean Model, version 1.0 (LICOM1.0), named LICOM-3DVM, has been developed using the three-dimensional variational data assimilation of mapped observation (3DVM), a 4DVar method newly proposed in the past two years. Two experiments with 12-year model integrations were designed to validate it. One is the assimilation run, called ASSM, which incorporated the analyzed weekly sea surface temperature (SST) fields from Reynolds and Smith (OISST) between 1990 and 2001 once a week by the LICOM-3DVM. The other is the control run without any assimilation, named CTL. ASSM shows that the simulated temperatures of the upper ocean (above 50 meters), especially the SST of equatorial Pacific, coincide with the Tropic Atmosphere Ocean (TAO) mooring data, the World Ocean Atlas 2001 (WOA01) data and the Met Office Hadley Centre’s sea ice and sea surface temperature (HadISST) data. It decreased the cold bias existing in CTL in the eastern Pacific and produced a Nino index that agrees with observation well. The validation results suggest that the LICOM-3DVM is able to effectively adjust the model results of the ocean temperature, although it’s hard to correct the subsurface results and it even makes them worse in some areas due to the incorporation of only surface data. Future development of the LICOM-3DVM is to include subsurface in situ observations and satellite observations to further improve model simulations.

Previous descriptions of how localized ensemble covariances can be incorporated into variational (VAR) data assimilation (DA) schemes provide few clues as to how this might be done in an efficient way. This article serves to remedy this hiatus in the literature by deriving a computationally efficient algorithm for using nonadaptively localized four-dimensional (4D) or three-dimensional (3D) ensemble covariances in variational DA. The algorithm provides computational advantages whenever (i) the localization function is a separable product of a function of the horizontal coordinate and a function of the vertical coordinate, (ii) and/or the localization length scale is much larger than the model grid spacing, (iii) and/or there are many variable types associated with each grid point, (iv) and/or 4D ensemble covariances are employed.

The use of Four-Dimensional variational (4D-Var) data assimilation technology in the context of sea dynamics problems, with a sensitivity analysis of model results to observation errors, is presented. The technology is applied to a numerical model of ocean circulation developed at the Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), with the use of the splitting method and complemented by 4D-Var data assimilation with covariance matrices of background and observation errors. The variational data assimilation involves iterative procedures to solve inverse problems so as to correct sea surface heat fluxes for the model under consideration. An algorithm is formulated to study the sensitivity of the model outputs, considered as output functions after assimilation, to the observation errors. The algorithm reveals the regions where the output function gradient is the largest for the average sea surface temperature (SST) in a selected area, obtained by assimilation. In the numerical experiments, a 4D variational problem of SST assimilation for the Baltic Sea area is solved.

Full Waveform Inversion (FWI) is a seismic imaging technique that utilizes the entire content of seismic trace. This method aims to minimizes differences between real and synthetic seismograms. FWI consists of three main steps: forward modelling to simulate synthetic seismograms, adjoint simulation to calculate the gradient misfit, and optimization to obtain an appropriate model perturbation. FWI is an extremely non-linear inversion technique and computationally intensive since it involves many model parameters. In the context of geophysics, finding parameter models that describe the structure of the earth’s subsurface is the main goal. Geophysical measurements in the field only obtain data that is influenced by the parameter model, in this case seismic recording data, therefore mathematical techniques are studied and applied to determine the correct parameter model from the data obtained, also known as the inversion process. In this study, main research area is on optimization step which include mathematical approach to find perturbation model effectively. The Newton method can be used in the optimization step of FWI. However, its implementation is not promising due to high computational cost in calculating Hessian matrix. To reduce computation cost, an identity matrix is chosen to replace the Hessian matrix, as it usually done in the steepest descent method. Since FWI has a high level of non-linearity, steepest descent method requires many iterations to minimize the objective function. Therefore, this study uses a conjugate gradient method to make the data misfit minimized more quickly. Apart from that, the Line search technique is used to find an appropriate step-length at each iteration on both methods. The FWI workflow is applied to the synthetic acoustic velocity model. We conduct FWI using steepest descent and conjugate gradient methods. The results show that the conjugate gradient method requires fewer iterations than the steepest descent method to minimize the objective function. FWI model result using conjugate gradient method is closer with the true model compared to steepest descent method, showed on misfit result and model image. By comparing these methods we can see that conjugate gradient method can be chosen for application on FWI optimization step for better result on seismic imaging. On the industry scale, better results provide less uncertainty on the model parameters that useful for energy exploration or engineering.

Large biases exist in real-time ENSO prediction, which can be attributed to uncertainties in initial conditions and model parameters. Previously, a 4D variational (4D-Var) data assimilation system was developed for an intermediate coupled model (ICM) and used to improve ENSO modeling through optimized initial conditions. In this paper, this system is further applied to optimize model parameters. In the ICM used, one important process for ENSO is related to the anomalous temperature of subsurface water entrained into the mixed layer (Te), which is empirically and explicitly related to sea level (SL) variation. The strength of the thermocline effect on SST (referred to simply as “the thermocline effect”) is represented by an introduced parameter, αTe. A numerical procedure is developed to optimize this model parameter through the 4D-Var assimilation of SST data in a twin experiment context with an idealized setting. Experiments having their initial condition optimized only, and having their initial condition plus this additional model parameter optimized, are compared. It is shown that ENSO evolution can be more effectively recovered by including the additional optimization of this parameter in ENSO modeling. The demonstrated feasibility of optimizing model parameters and initial conditions together through the 4D-Var method provides a modeling platform for ENSO studies. Further applications of the 4D-Var data assimilation system implemented in the ICM are also discussed.

<p>While fog can severely affect human activities (air, land and marine transportation), its forecast with current numerical weather prediction (NWP) models stays challenging, especially due to the lack of observations in the atmospheric boundary layer and the misrepresentation of non linear small scale processes. To improve knowledge on fog formation, evolution and dissipation, several instruments have been deployed during the SOFOG3D (SOuth west FOGs 3D experiment for processes study) experimental campaign to provide an unprecedent database of detailed 3D observations. In that context, a network of 8 ground-based microwave radiometers (MWR) provided continuous temperature profiling as well as integrated water vapor and liquid water path measurements during a 6 month period. Additionally, Martinet <em>et al</em> (2020) highlighted large temperature errors in the AROME-France (Application of Research to Operations at Mesoscale) NWP model background profiles during fog forecasts, leading to temperature differences up to 6 K when compared to tower measurements. Nevertheless, this study also demonstrate that the assimilation of MWR observations with a one dimensional variational data assimilation scheme could leads to improved initial conditions. To go further in that direction, MWR temperature profile observations from the SOFOG3D experiment have been added in the AROME-France operational data assimilation system, which uses a three dimensional variational algorithm (3D-Var) and climatological and homogeneous background error covariances (<strong>B</strong> matrix), to quantify the benefit on operational analyses and forecasts of several fog events. Then, the recently developped ensemble variational (EnVar) data assimilation system has been used to conduct new assimilation experiments. The main advantage of such method is to prescribe a fully flow dependent <strong>B</strong> matrix which is spatially and temporally coherent with the forecasted meteorological conditions. In consequences, it leads to more realistic increments. The results obtained with the different assimilation experiments will be presented. Firstly, a statistical analysis of the impact on the AROME-France analyses and short-range forecasts against conventional observations will be discussed. Secondly, specific SOFOG3D observations will be used to investigate the benefit on dedicated fog case studies.</p>