Part I. Introduction: 1. Inverse Problems and Data Assimilation in Earth Sciences Alik Ismail-Zadeh, Fabio Castelli, Dylan Jones and Sabrina Sanchez; 2. Emerging Directions in Geophysical Inversion Andrew P. Valentine and Malcolm Sambridge; 3. A Tutorial on Bayesian Data Assimilation Colin Grudzien and Marc Bocquet; 4. Third-Order Sensitivity Analysis, Uncertainty Quantification, Data Assimilation, Forward and Inverse Predictive Modelling for Large-Scale Systems Dan Cacuci; Part II. 'Fluid' Earth Applications from the Surface to the Space: 5. Data Assimilation of Seasonal Snow Manuela Girotto, Keith N. Musselman and Richard L. H. Essery; 6. Data Assimilation in Glaciology Mathieu Morlighem and Daniel Goldberg; 7. Data Assimilation in Hydrological Sciences Fabio Castelli; 8. Data Assimilation and Inverse Modelling of Atmospheric Trace Constituents Dylan Jones; 9. Data Assimilation of Volcanic Clouds: Recent Advances and Implications on Operational Forecasts Arnau Folch and Leonardo Mingari; 10. Data Assimilation in the Near-Earth Electron Radiation Environment Yuri Y. Shprits, Angelica M. Castillo, Nikita Aseev, Sebastian Cervantes, Ingo Michaelis, Irina Zhelavskaya, Artem Smirnov and Dedong Wang; Part III. 'Solid' Earth Applications from the Surface to the Core: 11. Trans-Dimensional Markov Chain Monte Carlo Methods Applied to Geochronology and Thermochronology Kerry Gallagher; 12. Inverse Problems in Lava Dynamics Alik Ismail-Zadeh, Alexander Korotkii, Oleg Melnik, Ilya Starodubtsev, Yulia Starodubtseva, Igor Tsepelev and Natalya Zeinalova; 13. Data Assimilation for Real-Time Shake-Mapping and Prediction of Ground Shaking in Earthquake Early Warning Mitsuyuki Hoshiba; 14. Global Seismic Tomography Using Time Domain Waveform Inversion Barbara Romanowicz; 15. Solving Larger Seismic Inverse Problems with Smarter Methods Lars Gebraad, Dirk-Philip van Herwaarden, Solvi Thrastarson and Andreas Fichtner; 16. Using Joint Inversion as a Hypothesis Testing Tool Max Moorkamp; 17. Crustal Structure and Moho Depth in the Tibetan Plateau from Inverse Modelling of Gravity Data Shuanggen Jin and S. B. Xuan; 18. Geodetic Inversions and Applications in Geodynamics Grigory Steblov and Irina Vladimirova; 19. Data Assimilation in Geodynamics: Methods and Applications Alik Ismail-Zadeh, Igor Tsepelev and Alexander Korotkii; 20. Geodynamic Data Assimilation: Techniques and Observables to Construct and Constrain Time-Dependent Earth Models Hans-Peter Bunge, Andre Horbach, Lorenzo Colli, Siavash Ghelichkhan, Berta Vilacís and Jorge N. Hayek; 21. Understanding and Predicting Geomagnetic Secular Variation via Data Assimilation Weijia Kuang, Kyle Gwirtz, Andrew Tangborn and Mathias Morzfeld; 22. Point-wise and Spectral Observations in Geomagnetic Data Assimilation, the Importance of Localization Sabrina Sanchez; Index.