Advanced new methods for multi-scale free surface regional ocean modelling with adjoint data assimilation

Lead Research Organisation: Imperial College London
Department Name: Earth Science and Engineering


The combined effect of population growth and industrialisation in the UK is such that coastal land areas are increasingly occupied by multiple user groups with diverse and competing needs (e.g. environmental, tourism, industrial). An important aspect of climate change is the increased likelihood of storms, and hence storm-surges and flooding, and this will have obvious impact upon low lying areas. There is thus an increased need to improve our capacity to predicti (especially over a wide range of spatial scales - a few meters to many kilometres) flooding. Improved modelling ability will inform policy makers, rescue services and scientists involved with ocean, climate change and risk reduction strategies. Data assimilation techniques are extremely valuable in compensating for lack of information about our oceans. Observed data is assimilated into models to produce an accurate estimate (in some optimal sense) of the state of the ocean. However, applications of efficient data assimilation approaches (e.g., variational data assimilation) are hampered by two major difficulties: the often complex code (implementation and maintainability) required; and the high computational costs. To address these issues, the proposed work will improve the existing models by using: 1) a newly developed data assimilation formulation to dramatically reduce the code complexity and increase maintainability; 2) a new highly stable and accurate wetting and drying method capable of resolving multi-scale physics and uniquely designed for use with a next generation ocean model; 3) model reduction in which large-scale models are reduced down to a few hundred unknowns so that the resulting models are orders of magnitude faster than the original model. Our overall aim is the accurate prediction of free surface dominated flows in coastal regions. Prediction will be achieved by developing a variational data assimilation (in the content of the time dependent problems solved here) framework within our advanced adaptive mesh ocean model. This framework will be capable of quantifying the effect of model uncertainties, performing sensitivity analysis, and capturing abruptly changing fields such as wetting and drying fronts in free surface dominated regional flows. This will pave the way towards an open source, community, next generation regional ocean model with multi-scale adaptive finite element meshing features and predictive capability. The overall deliverable will be a model capable of resolving free surface dominated flows from ocean to estuary (and smaller scale) scale. The proposed combination of recently developed techniques is the only feasible way of resolving these demanding multi-scale flows. The proposed research is also expected to have a substantial impact on the future development of operational implementation of variational data assimilation in both meteorology and oceanography. The reduction in computational effort and memory requirements will render data assimilation a more affordable research and operational tool. Society as a whole will benefit from this research through improved prediction of multi-scale coastal flows, especially the prediction of storm flooding. In particular, government, regulatory bodies and stakeholder, companies/industries, meteorology, oceanography communities and institutes would benefit from the new technologies that could be used for prediction and impact assessment of natural disasters, pollution and rapid emergency response.

Planned Impact

This project delivers a step-change improvement in the ability of models to predict flooding and pollution dispersal in coastal environments. This work will impact upon the following: (1) Consulting companies: with an interest in using the new technologies to support decision making and prediction and impact assessment of natural disasters, pollution and development of rapid emergency response strategies. The new technologies could help inform the design of mitigation strategies for flooding events associated with storm surges and tsunamis (e.g. identifying the origin of the tsunami generation by assimilating tide gauge data into models). (2) Engineer: would benefit from availability of tools that could be used to optimise design in coastal engineering (e.g. flooding defence, coastal structures). (3) Observational data centres/companies: would benefit from the ability to assimilate data into models so as to optimise placement of monitoring/collecting stations (significantly reducing the cost and time) e.g. the maxima of the reduced order model basis functions indicates where data needs to be collected from. (4) Designers in companies/industries: would benefit from the optimisation control approach used in design optimisation, in for example, shapes of industrial devices, cars and aeroplanes. (5) Power companies: involved in tidal barrage and nuclear projects. (6) Environmental agency/industries: would benefit from improved predictability of modelling dispersal of pollutants and industrial effluent in shallow-water regimes, furthermore this work could be also used to assess strategies used to reduce pollutant emissions and toxic releases in urban environments. (7) Government security bodies: the reduced order approached proposed here could provide a tool in the design and implementation of effective emergency response actions in the case of toxic pollutant release (for example,in city centres and the London underground). (8) Government, regulatory bodies and stakeholder: (Met Office, DEFRA/EA) could use this technology to inform policy makers, rescue services and scientists involved with ocean, climate change, risk reduction strategies for the UK. The investigators of this proposal have considerable experience in knowledge exchange and commercial exploitation of their work. They have many past and on-going collaborations with industrial companies. This will help us exploit the project results, with an expected high national and international impact. Proudman Oceanographic Laboratory (POL) and HR Wallingford will provide the corresponding observed data and test/apply the Imperial College Ocean Model (in Mersey Estuary - POL, Severn and Teignmouth Estuaries - HR Wallingford). In addition, HR Wallingford will support a PhD student for 3 years to work as part of this programme; Istituto Ambiente Marinoe Costiero (IAMC, Italy) will consider the re-application to the Venice lagoon; SeaZone will provide high-resolution bathymetric data. Planned activities to ensure good engagement and communication with beneficiaries include: i) trainging events on the science of this proposal and the resulting open source model: organising workshops,seminars, Summer school and postgraduate courses; ii) conference presentations, publications, reports etc; iii) collaboration with universities and industries; and iv) Improved web based presence.


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Ardjmandpour N (2014) Reduced order borehole induction modelling in International Journal of Computational Fluid Dynamics

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Buchan A (2013) A POD reduced-order model for eigenvalue problems with application to reactor physics in International Journal for Numerical Methods in Engineering

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Che Z (2014) An ensemble method for sensor optimisation applied to falling liquid films in International Journal of Multiphase Flow

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Du J (2013) POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow in Computers & Mathematics with Applications

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Du J (2016) Ensemble data assimilation applied to an adaptive mesh ocean model in International Journal for Numerical Methods in Fluids

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Fang F (2017) An efficient goal-based reduced order model approach for targeted adaptive observations in International Journal for Numerical Methods in Fluids

Description 1. Development of a 3D reduced order model within Fluidity-ICOM (see publications [1, 2]

A POD based reduced order model has been developed for Fluidity-ICOM, an unstructured fluid model, which can simultaneously resolve both small and large scale ocean flows while smoothly varying the resolution and conforming to complex coastlines and bathymetry. This discrete POD model retains the advanced numerical techniques as Fluidity-ICOM continues to be developed.

The new model has been applied to the flow past a sphere and 3D air pollution cases [1, 2]

2. New approaches developed for stabilisation of reduced order modelling (see publications [3, 4]

Development: The traditional POD/Galerkin finite element model lacks stability and spurious oscillations can degrade the reduced order solution for flows with high Reynolds numbers. In this work, a new nonlinear Petrov-Galerkin approach has been developed for proper orthogonal decomposition (POD) reduced order modelling (ROM) of the Navier-Stokes equations. The new method is based on the use of the cosine rule between the advection direction in Cartesian space-time and the direction of the gradient of the solution. The contribution of this work lies in applying this new non-linear Petrov-Galerkin method to the reduced order Navier-Stokes equations, and thus improving the stability of ROM results without tuning parameters.

The results of numerical tests are presented for a wind driven 2D gyre and the flow past a cylinder, which are simulated using the unstructured mesh finite element CFD model in order to illustrate the numerical performance of the method. The numerical results obtained show that the newly proposed POD Petrov-Galerkin method can provide more accurate and stable results than the POD Bubnov-Galerkin method.

3. New approaches developed for the treatment of the equation's nonlinear operators [see publication [5])

Due to the high non-linearities of the 3-D Bubnov-Galerkin Navier-Stokes equation, the computational complexity of the reduced model still depends on dimension of the full Navier-Stokes discretisation. In this project, a new method (named residual DEIM) has been developed for treatment of the equation's non-linear operator. This method provides accurate simulations within an efficient framework. The method itself is a hybrid of two existing approaches, namely the quadratic expansion method and the Discrete Empirical Interpolation Method (DEIM), that have already been developed to treat non-linear operators within reduced order models. In addition to the treatment of the non-linear operator the POD model is stabilized using the Petrov-Galerkin method developed here. This adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions.

The capabilities of this new approach are demonstrated by solving the incompressible Navier-Stokes equations for simulating a flow past a cylinder and gyre problems. Comparisons are made with other treatments of non-linear operators, and these show the new method to provide significant improvements in the solution's accuracy while the computational times were reduced in comparison to the full model calculations, where for the larger meshes tested the CPU costs were reduced by up to 98%.

4. Development of a reduced order 4D variational (adjoint) model (see publication [6])

A new reduced order adjoint model has been developed within Fluidiity-ICOM. In this work, the reduced adjoint model is derived directly from the discretized reduced forward model. The whole optimization procedure is undertaken completely in reduced space. The computational cost for the 4D variational data assimilation is significantly reduced by decreasing the dimensional size of the control space, in both the forward and adjoint models. Computational efficiency is further enhanced since both the reduced forward and adjoint models are constructed by a series of time-independent sub-matrices. The reduced forward and adjoint models can be run repeatedly with negligible computational costs.

The new POD model is validated using the Munk gyre flow test case, where it inverts for initial conditions. The optimized velocity fields exhibit overall good agreement with those generated by the full model.

The new adjoint model is further applied for optimisation of sensor locations. The goal function is used to optimally locate the grid of sensors across a deployable field. Data from the grid is continuously assimilated into the model which is then able to predict the goal function with the best possible accuracy.

5. Publications

1) J. Du, F. Fang, C.C. Pain, I.M. Navon, J. Zhu, D. Ham. POD reduced-order unstructured mesh modelling applied to 2D and 3D fluid flow. ''Computers & Mathematics with Applications'', 65, (2013), 362-379.

2) F. Fang, D.Pavlidis, C.C.Pain, A.G.Buchan, I.M.Navon, Reduced order modelling of an unstructured mesh air pollution model and application in 2D/3D urban street canyons, submitted to Atmospheric Environment, 2013.

3) F. Fang, C.C. Pain, I.M. Navon, A.H. Elsheikh, J. Du, D. Xiao. Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods, J. Comput. Phys., 234, (2013), 540-559.

4) D. Xiao., F. Fang, J. Du, C.C. Pain, I.M. Navon, A.G Buchan, A.H. Elsheikh, G Hu. Non-Linear Petrov-Galerkin Methods for Reduced Order Modelling of the Navier-Stokes Equations using a Mixed Finite Element Pair. Computer Methods in Applied Mechanics and Engineering, 255, (2013), 147-157.

5) F. Fang, C.C.Pain, I.M.Navon, etc. An efficient adjoint approach for targeting adaptive observation in data assimilation, in preparation.

6) F. Fang, C.C. Pain, I.M. Navon, D.G.Cacuci, X.Chen. The independent set perturbation method for efficient computation of sensitivities with applications to data assimilation and a finite element shallow water model, Computer and Fluids, 76, (2013), 33-49.

7) A. G. Buchan, C.C. Pain, F. Fang, I. M. Navon. A POD reduced order model for eigenvalue problems with application to Reactor Physics. International Journal for Numerical Methods in Engineering. 95, (2013), 1011-1032.

8) J. Du, I.M. Navon, J. Zhu, F. Fang, A.K. Alekseev. Reduced order modeling based on POD of a parabolized Navier-Stokes equations model II: trust region POD 4-D VAR Data Assimilation. ''Computers & Mathematics with Applications'', 65, (2013), 380-394.

9) H. ELSheikh, C.C. Pain, F. Fang, J.L.M.A. Gomes, I.M. Navon. Parameter estimation of subsurface flow models using iterative regularized ensemble Kalman filter. ''Stochastic Environmental Research and Risk Assessment'', 27, (2013), 877-897.

10) Hossen, M.J, Navon I.M, Fang F, A Penalized 4-D Var data assimilation method for reducing forecast error related to adaptive observations, International Journal for Numerical Methods in Fluids, 70(10), (2012), 1207-122.

Training Events:

• 3-day training course is held every year in Nov. This training course is designed to get people started with Fluidity. It starts by covering the mathematical and numerical aspects of the finite element method, then moves onto Fluidity itself, including: building Fluidity; setting up simulations and running them; and visualisation. The final day is dedicated to some example simulations and hands-on training.

• The Fluidity training courses were held in Tsinghua Univ., the Institute of Atmospheric Physics, ZheJiang Univ. and Wuhan Univ, in China in 2011 and 2013.

International Collaboration activities:

• Collaboration with Florida state university: 1) Prof. Navon visited us once a year; 2) 10 journal papers have been published.

• Collaboration with Chinese universities:

1) the workshops and training courses were held in Tsinghua Univ., the Institute of Atmospheric Physics (IAP), ZheJiang Univ. and Wuhan Univ.

2) 7 joint journal papers with IAP have been published;

3) Joint research with IAP: Ensemble Kalman data assimilation method has been developed within Fluidity, and Fluidity has been applied to South China Sea.
Exploitation Route This work will be of use to anyone who is concerned with ocean modelling and data assimilation, natural hazard (flooding, Tsunami etc.) management. The innovative approaches to data assimilation will lead the way to improved operational model/service in both meteorology and oceanography.

A number of companies have placed contracts with us, including: British Energy, AWE, Rolls-Royce, Babcock and Wilcox (USA), JAEA (Japan) and IRSN (France). Although the end-product (technical reports and software licensing) had an excellent acceptance, issues regarding the lack of the software's user-friendliness and documentation were raised and need to be tackled.

Therefore, as a strategy to enable the academic and commercial software dissemination we need to:

(a) Further develop a new user-friendly front-end;

(b) Include comprehensive documentation of the modules in Fluidity;

(c) Develop diagnostic tools specifically designed for industries;

(d) Enable the use of third-party mesh generators;

(e) Further develop model and software quality assurance procedures focused on the industry requirements. 1. Collaborate with industrial companies and other stakeholders will help us exploit the project results, with an expected high national and international impact.

2. Provide training courses (we have organised a number of traning courses in the last three years).

3. Publish journal papers (10 papers on reduced order modelling have been published during this research project).

4. Present our results in international conferences (we have given talks at SIAM, APS, AGOS, AGU, Ocean meeting, ICIAM, universities etc.).

5. Imperial Innovations has experience of many software spin-outs especially in the processing industries. This expertise will be used to negotiate the type of flexible agreements that would be required for research consultancy and collaborative work, or expanding the commercial use of our tools.
Sectors Energy,Environment

Description Collaboration with FSU: Reduced order modelling and data assimilation 
Organisation Florida State University
Country United States of America 
Sector Academic/University 
PI Contribution The major research areas: 1. 4D variational data assimilation. 2. Numerical modeling. 3. Reduced order modelling. 3. Inversion problems. 1. 10 joint journal papers have been published in the last three years. 2. Prof, Navon visits us once a year. 3. Prof. Navon trains our staff and PhD students during his visit every year. 4, Prof. Navon provides postgraduate tuition/lectures. 5. We apply joint research fund in the UK and USA.
Start Year 2002
Description Collaboration with The Institute of Atmospheric Physics (IAP): Application of an unstructured mesh FE ocean model (Fluidity-ICOM) to South China Sea and Data assimilation 
Organisation Chinese Academy of Sciences
Department Institute of Atmospheric Physics (IAP)
Country China, People's Republic of 
Sector Academic/University 
PI Contribution Major collaborative research areas: 1. Application of an unstructured mesh FE ocean model (Fluidity) to South China Sea. 2. Deveolopment of an Ensemble Kalman Filter data assimilation system within Fluidity-ICOM. 3. Reduced order modelling and applications. 4. Air pollution. 5. Fluid-soild coupling. 1. 8 joint journal papers have been published. 2. A EnKF data assimilation system has been developed within Fluidity-ICOM. 3. Two joint meetings were held in 2010 (Beijing) and 2011 (London). 4. A workshop took place in Beijing in Aug. 2012. 5. The 1st FLUIDITY/FEMDEM training course was held in Beijing in March, 2013. 6. One PDRA and two PhD students work with AMCG annually.
Start Year 2011