This book describes the methods and numerical approaches for data assimilation in geodynamical models and presents several applications of the described methodology in relevant case studies. The book starts with a brief overview of the basic principles in data-driven geodynamic modelling, inverse problems, and data assimilation methods, which is then followed by methodological chapters on backward advection, variational (or adjoint), and quasi-reversibility methods. The chapters are accompanied by case studies presenting the applicability of the methods for solving geodynamic problems; namely, mantle plume evolution; lithosphere dynamics in and beneath two distinct geological domains ¿ the south-eastern Carpathian Mountains and the Japanese Islands; salt diapirism in sedimentary basins; and volcanic lava flow.

Applications of data-driven modelling are of interest to the industry and to experts dealing with geohazards and risk mitigation. Explanation of the sedimentary basin evolution complicated by deformations due to salt tectonics can help in oil and gas exploration; better understanding of the stress-strain evolution in the past and stress localization in the present can provide an insight into large earthquake preparation processes; volcanic lava flow assessments can advise on risk mitigation in the populated areas. The book is an essential tool for advanced courses on data assimilation and numerical modelling in geodynamics.

Alik Ismail-Zadeh is a Chief Scientist / Research Professor of the Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences (RusAS) at Moscow, Russia, and a Senior Scientist at the Institute of Applied Geosciences, Karlsruhe Institute of Technology, Germany. He graduated from the Baku State and Lomonossov Moscow State Universities (mathematics and physics) and received his PhD and Doctor of Science degrees (in geophysics) from RusAS. He has been lecturing on computational geodynamics at the University of Karlsruhe, Abdus Salam International Center for Theoretical Physics in Trieste, and Moscow State University of Oil and Gas, and working as visiting scholar/professor in the universities of California (UCLA), Cambridge, Paris (IPGP), Stockholm (KTH), Tokyo, Trieste, and Uppsala. His scientific interests cover studies of the crust and mantle dynamics, basin evolution, salt tectonics, and seismic hazard through theoretical analysis and numerical modelling. He is an author of over 100 peer-reviewed papers and four books. Alik Ismail-Zadeh is an honorary fellow of the Royal Astronomical Society and has been awarded the Academia Europaea Medal, AGU International Award, and several prestigious fellowships including Alexander von Humboldt, Royal Society of London, Royal Swedish Academy of Sciences, and Russian President.

Alexander Korotkii is Head of the Department of Applied Problems of the Institute of Mathematics and Mechanics, Russian Academy of Sciences at Yekaterinburg, Russia. He graduated from Ural State University (mathematics and mechanics) and received his PhD and Doctor of Science degrees (in mathematics) from RusAS. He is also Professor in Mathematics at the Chair of Numerical Mathematics of the Ural Federal University and teach on computational methods and mathematical modeling. His scientific interests cover studies of partial differential equations, optimal control theory, systems of distributed parameters with uncertainties and conflicts, direct and inverse problems in fluid dynamics. He is an author and co-author of 80 research papers.

Igor Tsepelev is a Senior Scientist of the Department of Applied Problems of the Institute of Mathematics and Mechanics, RusAS at Yekaterinburg, Russia. He graduated from the Ural State University (mathematics and mechanics) and received his PhD in mathematics (differential equations and inverse problems) from RusAS. His scientific interests cover studies of inverse ill-posed problems, development of numerical methods for solving systems of partial differential equations and algorithms for parallel computing, and mathematical and numerical modeling in geophysics. He is an author and co-author of more than 50 research papers. 

1. Introduction

1.1 Inverse problems in geodynamics

1.2 Forward and backward modelling and source of errors

1.3 Data assimilation methods

2. Backward advection method and its application to salt tectonics

2.1 Basic idea of the backward advection (BAD) method

2.2 Modelling of salt diapirism

2.3 Mathematical statement

2.4 Solution method

2.5 Forward and backward model results

3. Variational method and its application to mantle plume evolution

3.1 Basic idea of the variational (VAR) method

3.2 Mathematical statement

3.3 Objective functional

3.4 Adjoint problem

3.5 Solution method

3.6 Restoration of mantle plumes

3.7 Challenges in variational data assimilation

4. Application of the VAR method to volcanic lava dynamics

4.1 Volcanic lava flow

4.2 Reconstruction of volcanic lava properties

4.3 Mathematical statement

4.4 Minimisation problem

4.5 Adjoint problem

4.6 Numerical approach

4.7 Model results and discussion

5. Quasi-reversibility method and its applications

5.1 Basic idea of the quasi-reversibility (QRV) method

5.2 Mathematical statement

5.3 Optimisation problem and numerical approach

5.4 Restoration of mantle plumes

5.5 Restoration of descending lithosphere evolution

6. Application of the QRV method to reconstruction of plate subduction

6.1 Plate subduction beneath the Japanese islands

6.2 Mathematical statement

6.3 Input data: Seismic temperature model

6.4 Boundary conditions

6.5 Rheological model

6.6 Numerical approach

6.7 Model results

6.8 Data uncertainties

7. Comparison of data assimilation methods for mantle convection models