Model Reduced Variational Data Assimilation for Shallow Water Flow Models
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op maandag 31 januari 2011 om 10.00 uur
door
Muhammad Umer ALTAF, Master of Science in Mathematics, Quaid-e-Azam University, Pakistan geboren te Rawalpindi, Pakistan.
Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. A. W. Heemink
Copromotor: Dr. ir. M. Verlaan
Samenstelling promotiecommissie:
Rector Magnificus Prof. dr. ir. A. W. Heemink, Dr. ir. M. Verlaan, Prof. dr. ir. A. Mynett, Prof. dr. ir. C. Vuik, Prof. dr. ir. E. Deleersnijder, Prof. dr. ir. J.H. van Schuppen, Dr. H. Madsen,
voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft, copromotor UNESCO-IHE/Technische Universiteit Delft Technische Universiteit Delft Universite Catholique de Louvain, Belgium Technische Universiteit Delft DHI Group, Denmark
This thesis has been completed in partial fulfillment of the requirements of Delft University of Technology (Delft, The Netherlands) for the award of the Ph.D. degree. The research described in this thesis was carried out in the Delft Institute of Applied Mathematics, Delft University of Technology.
Published and distributed by: Muhammad Umer ALTAF E-mail:
[email protected]
ISBN # 978-90-8570-718-9
Copyright © 2010 by Muhammad Umer ALTAF All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission of the author. Printed in The Netherlands: by W¨ohmann Print Service
Contents
1 Introduction 1.1 Storm surge forecasting . . . . . . . . . . 1.2 Data assimilation and model calibration . 1.3 Model reduction . . . . . . . . . . . . . . 1.3.1 Proper Orthogonal Decomposition 1.4 Motivation and overview . . . . . . . . .
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2 The Dutch Continental Shelf Model 2.1 The model equations . . . . . . . . . . . . . . . . . . 2.2 Numerical approximation . . . . . . . . . . . . . . . . 2.2.1 State space model . . . . . . . . . . . . . . . . 2.3 Adjoint method . . . . . . . . . . . . . . . . . . . . . 2.3.1 Procedural flow for the classical adjoint method 2.3.2 The LBFGS method . . . . . . . . . . . . . . 2.4 An overview on DCSM calibration . . . . . . . . . . . 2.5 Operational forecasts using DCSM . . . . . . . . . . .
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3 POD and BPOD within variational data assimilation 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model reduction . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Proper Orthogonal Decomposition . . . . . . . . 3.2.2 Balanced Truncation . . . . . . . . . . . . . . . 3.2.3 Balanced POD . . . . . . . . . . . . . . . . . . 3.3 Inverse modeling using reduced models . . . . . . . . . 3.3.1 Linearization and reduced model formulation . . 3.3.2 Collection of the snapshots and the reduced basis 3.3.3 Approximate objective function and its adjoint . 3.4 Numerical experiments . . . . . . . . . . . . . . . . . . 3.4.1 The Model . . . . . . . . . . . . . . . . . . . . 3.4.2 Experiment 1 . . . . . . . . . . . . . . . . . . . 3.4.3 Experiment 2 . . . . . . . . . . . . . . . . . . . iii
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Contents 3.5
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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inverse shallow water flow modeling using model reduction 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Inverse modeling . . . . . . . . . . . . . . . . . . . . . 4.3 Reduced order modeling . . . . . . . . . . . . . . . . . 4.3.1 Proper Orthogonal Decomposition . . . . . . . . 4.4 Inverse modeling using reduced models . . . . . . . . . 4.4.1 Linearization and reduced Basis . . . . . . . . . 4.4.2 Reduced model formulation . . . . . . . . . . . 4.4.3 Approximate objective function and its adjoint . 4.4.4 Procedural flow with reduced model . . . . . . . 4.4.5 Computational cost . . . . . . . . . . . . . . . . 4.5 Application: The Dutch Continental Shelf Model . . . . 4.5.1 Estimation of depth . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
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Case Study: Identification of uncertain parameters in a large scale tidal model 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The new Dutch continental shelf model . . . . . . . . . . . . . . . . 5.2.1 Model computational grid of the new model . . . . . . . . . . 5.2.2 Model bathymetry and bottom roughness . . . . . . . . . . . 5.2.3 Model boundary conditions . . . . . . . . . . . . . . . . . . 5.2.4 Initial conditions and computational time . . . . . . . . . . . 5.3 Inverse modeling using POD . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Linearization and reduced model formulation . . . . . . . . . 5.3.2 Collection of the snapshots and POD basis . . . . . . . . . . 5.3.3 Approximate objective function and its adjoint . . . . . . . . 5.3.4 Workflow with POD algorithm . . . . . . . . . . . . . . . . . 5.3.5 Convergence criterion for inner and outer iterations . . . . . . 5.3.6 Computational efficiency of the algorithm . . . . . . . . . . . 5.4 Parameter identification for the new DCSM . . . . . . . . . . . . . . 5.4.1 Measurement data . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 The calibration and validation datasets and time period . . . . 5.4.3 Time and frequency domain analysis . . . . . . . . . . . . . . 5.4.4 Ensemble generation . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Model calibration . . . . . . . . . . . . . . . . . . . . . . . . 5.4.7 Discussion on results . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter Estimation using Simultaneous Perturbation 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Parameter estimation using SPSA . . . . . . . . . . . . . . . . . . . 6.2.1 Choice of al and cl . . . . . . . . . . . . . . . . . . . . . . .
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Contents
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6.2.2 Average stochastic gradient Numerical experiments . . . . . . . 6.3.1 Experiment 1 . . . . . . . . 6.3.2 Experiment 2 . . . . . . . . Conclusions . . . . . . . . . . . . .
7 Conclusions
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104 105 105 110 114 117
Appendices A The Reduced Adjoint State
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Bibliography
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Summary
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Samenvatting
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Acknowledgements
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Curriculum Vitae
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List of Tables
2.1
3.1 3.2 3.3 3.4
5.1 5.2 5.3 5.4 5.5 5.6
6.1 6.2
Warning and alarm levels used by SVSD (level relative to normal Amsterdam level (NAP)) . . . . . . . . . . . . . . . . . . . . . . . . . . The results of the estimation of two parameters γ1 and γ2 . Here α and β are the number of inner and outer iterations respectively. . . . . . . Results of the estimation of two parameters γ1 and γ2 with the reduced model that captured 99.99% of the relative energy. . . . . . . . . . . . Results of the estimation of four parameters γ1 , γ2 , γ3 and γ4 with the POD based estimation procedure. . . . . . . . . . . . . . . . . . . . . RRMSE in the forward simulation with different model reduction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RMSE results for the minimization process with the 97% relative energy after 1 st and 2nd outer iteration (Experiment 1) . . . . . . . . . . RMSE results for the minimization process with the 90% relative energy after 1 st and 2nd outer iteration (Experiment 1) . . . . . . . . . . Results after 2nd outer iteration obtained by generating new ensemble and using the same ensemble as in 1 st outer iteration (Experiment 1). . RMSE results for the minimization process with the 90% relative energy after 3rd and 4th outer iterations (Experiment 2) . . . . . . . . . RMSE results for the minimization process with the 97% relative energy after 5th outer iteration (Experiment 3) . . . . . . . . . . . . . . Computational costs of the calibration experiments after each outer iteration β of the minimization process . . . . . . . . . . . . . . . . .
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85 86 87 89 91 96
Comparison of estimated parameters to true parameters for the twin experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 RMSE results for the minimization process after 5th , 10th , 15th and 20th iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
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List of Figures
1.1 1.2
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Eastern Scheldt barrier at North Sea during storm . . . . . . . . . . . The Eastern Scheldt storm surge barrier, or Oosterschelde. (photo by Rob Broek/iStockPhoto.com) . . . . . . . . . . . . . . . . . . . . . . Storm surge barrier at the New Waterwey. (photo taken from Microsoft Encarta Online Encyclopaedia 2008) . . . . . . . . . . . . . . The inverse structure of the data assimilation problems . . . . . . . .
2.1 2.2 2.3
Area of the operational DCSM. . . . . . . . . . . . . . . . . . . . . . The computational grid . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of calibration process using adjoint method [2] . . . .
3.1
Flowcharts of the parameter estimation procedure with the classical adjoint method (left) and the reduced model approaches (middle (POD) and right (BPOD)) . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The 2D pollution model domain; The locations of the selected data points (circles) and straight line divides the model domain in two regions one for each estimated parameter . . . . . . . . . . . . . . . . . The POD modes captured energy for an ensemble of 100 snapshot vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The value of the objective function J at successive outer iterations for reduced models with different relative energy attained . . . . . . . . . The locations of the selected data points (circles) and straight line divides the model domain in two regions one for each estimated parameter The locations of the selected data points (circles) and the four subdomains used used in case 3 to estimate four parameters . . . . . . . . . The POD modes captured energy for an ensemble of 200 snapshots in the outer iterations a) β1 b) β2 . . . . . . . . . . . . . . . . . . . . . . The 2D pollution model domain; The locations of the selected data points (circles) and straight line divides the model domain in two regions one for each estimated parameter . . . . . . . . . . . . . . . . .
1.3
3.2 3.3
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List of Figures 3.10 The successive outer iterations β of the objective function (J) for the POD and the BPOD based reduced order models of order
