Data assimilation (DA) is a collective term for mathematical methods that consistently combine models with observations. A great advantage of DA is that the information contained in the observations can be used to derive knowledge about unobserved, e.g., unobservable, quantities. This can be achieved by exploiting physical or statistical links between observed and unobserved variables (Fig. 1). 

DA in our section focuses mainly on geodetic observations, e.g., Earth rotation, satellite gravimetry and satellite altimetry [Saynisch et al., 2011a, 2015]. In addition, new technologies are tested in so called Observing System Simulation Experiments (OSSE) for their potential benefit to the respective research community, as e.g., the newly emerging GNSS-Reflectometry technology. 

In general, the observations are assimilated with state-of-the-art numerical models of Earth’s sub-systems, e.g., atmosphere, oceans, mantle and cryosphere [Neef and Matthes, 2012, Irrgang et al., 2017, Bernales et al., 2017].

Prior to a successful assimilation, realistic error budgets have to be derived. On the one hand, these budgets describe realistic statistics of the observations ,e.g., from GRACE [Dobslaw et al., 2016] or GNSS-R [Semmling et al., 2016]. On the other hand, these budgets have to describe the uncertainties and sensitivities of the used numerical models, see Fig. 2 [e.g., Zhang et al., 2017, Dill et al., 2015].

For the model error estimation numerically expensive ensemble calculations are required [e.g., Irrgang et al., 2016]. As part of benchmark experiments, these ensembles are ideally based on very different models [Saynisch et al., 2017, Sachl et al., 2017]. The ensemble information is subsequently used in state-of-the-art Ensemble Kalman Filters [Irrgang et al., 2017]. Due to the typical high-dimensionality of models of Earth System components, the ensemble generation and the Kalman filters have to operate on small yet dynamically optimal subspaces [e.g., Nerger et al., 2005].

In addition to Kalman filter techniques, Section 1.3 also uses a range of variational or adjoint techniques [Saynisch et al., 2011b, 2015]. Based on the specific research question, the models and the formulation of the assimilation method can be regional or global, statistical or variational. In particular cases the same observations are assimilated with both adjoint and Kalman-filter techniques to increase the robustness of the results [cf., Saynisch and Thomas, 2012, Saynisch et al., 2011b]. 

As part of the international GEROS-ISS project, we could for the first time demonstrate the information gain of GNSS-Reflectometry measurements for the oceanographic community and could give specific recommendations for undecided questions of observations density and precision [Saynisch et al., 2015, Wickert et al., 2016]. Within the SMART-Cables project, which aims to fit telecommunication cables with oceanographic sensors, a cable deployment strategy could be proposed that bases on the respective cable’s assimilation impact. As part of the Dynamic Earth-SPP, pioneering work is done in studying the potential of the currently ongoing satellite magnetometer mission Swarm for oceanic assimilation purposes [Irrgang et al., 2017, Saynisch et al., 2017].

• Assimilation of oceanic magnetic signals
• Estimating global ocean heat content from tidal magnetic signals with machine learning
• Assimilation of GNSS-R based satellite altimetry data with a regional high resolution ocean modell
• Assimilation of SMART Cables
• Assimilation of Earth Rotation and the Ocean's Circulation (EROC)
• Assimilation of Co-Amplitudes in the Barotopic Tidal Modul
• Assimilation of satellite gravimetry observations into a global ocean circulation model
• SCENIC-Project

Schachtschneider, R., Saynisch-Wagner, J., Klemann, V., Bagge, M., Thomas, M. (2022): An approach for constraining mantle viscosities through assimilation of palaeo sea level data into a glacial isostatic adjustment model. - Nonlinear Processes in Geophysics, 29, 1, 53-75. https://doi.org/10.5194/npg-29-53-2022

Dill, R., Saynisch-Wagner, J., Irrgang, C., Thomas, M. (2021): Improving atmospheric angular momentum forecasts by machine learning. - Earth and Space Science, 8, 12, e2021EA002070. https://doi.org/10.1029/2021EA002070

Drinkorn, C., Saynisch-Wagner, J., Uenzelmann-Neben, G., Thomas, M. (2021): Decadal climate sensitivity of contouritic sedimentation in a dynamically coupled ice-ocean-sediment model of the North Atlantic. - Palaeogeography Palaeoclimatology Palaeoecology, 572, 110391. https://doi.org/10.1016/j.palaeo.2021.110391

Saynisch-Wagner, J., Baerenzung, J., Irrgang, C., Hornschild, A., Thomas, M. (2021): Tide induced magnetic signals and their errors derived from CHAMP and Swarm satellite magnetometer observations. - Earth Planets and Space, 73, 234. https://doi.org/10.1186/s40623-021-01557-3 

Irrgang, C., Dill, R., Boergens, E., Saynisch-Wagner, J., Thomas, M. (2020): Self-validating deep learning for recovering terrestrial water storage from gravity and altimetry measurements. - Geophysical Research Letters, 47, 17, e2020GL089258. https://doi.org/10.1029/2020GL089258

J. Bernales, I. Rogozhina, and M. Thomas. Melting and freezing under antarctic ice shelves from a combination of ice-sheet modelling and observations. J. Glaciology, 63(240):731–744, 2017.

R. Dill, V. Klemann, Z. Martinec, and M. Tesauro. Applying local green’s func- tions to study the influence of the crustal structure on hydrological loading displacements. J. Geodyn., 88(Supplement C):14–22, 2015.

H. Dobslaw, I. Bergmann-Wolf, E. Forootan, C. Dahle, T. Mayer-Gürr, J. Kusche, and F. Flechtner. Modeling of present-day atmosphere and ocean non-tidal de-aliasing errors for future gravity mission simulations. J. Geodesy, 90(5): 423–436, 2016.

C. Irrgang, J. Saynisch, and M. Thomas. Utilizing oceanic electromagnetic induction to constrain an ocean general circulation model: A data assimilation twin experiment. J. Adv. Model. Earth Sys., 9(3):1703–1720, 2017.

C. Irrgang, J. Saynisch, and M. Thomas. Ensemble simulations of the magnetic field induced by global ocean circulation: Estimating the uncertainty. J. Geophys. Res, 121(3):1866–1880, 2016.

L. J. Neef and K. Matthes. Comparison of Earth rotation excitation in data- constrained and unconstrained atmosphere models. J. Geophys. Res., 117 (D02107):1–17, 2012.

L. Nerger, W. Hiller, and J. Schröter. A comparison of error subspace Kalman filters. Tellus Ser. A: Dyn. Meteorol. Oceanol., 57(5):715–735, 2005.

L. Sachl, Z. Martinec, J. Velimsk, A. Grayver, C. Irrgang, A. Kuvshinov, J. Petereit, J. Saynisch, D. Einspigel, and N. R. Schnepf. Benchmark study of global EM induction codes forced by ocean circulation electric currents. Geo- phys. J. Int., submitted:1–19, 2017.

J. Saynisch and M. Thomas. Ensemble Kalman-Filtering of Earth rotation observations with a global ocean model. J. Geodyn., 62:24–29, 2012.

J. Saynisch, M. Wenzel, and J. Schröter. Assimilation of Earth rotation parameters into a global ocean model: length of day excitation. J. Geodesy, 85(2): 67–73, 2011a.

J. Saynisch, M. Wenzel, and J. Schröter. Assimilation of Earth rotation pa- rameters into a global ocean model: excitation of polar motion. Nonlinear Process. Geophys., 18(5):581–585, 2011b.

J. Saynisch, I. Bergmann-Wolf, and M. Thomas. Assimilation of GRACE de- rived oceanic mass distributions with a global ocean circulation model. J. Geodesy, 89(2):121–139, 2015.

J. Saynisch, J. Petereit, C. Irrgang, and M. Thomas. Impact of oceanic warm- ing on electromagnetic oceanic tidal signals: A CMIP5 climate model-based sensitivity study. Geophys. Res. Lett., 44(10):4994–5000, 2017.

J. Saynisch, M. Semmling, J. Wickert, and M. Thomas. Potential of space-borne GNSS reflectometry to constrain simulations of the ocean circulation. Ocean Dyn., 65(11):1441–1460, 2015.

A. M. Semmling, V. Leister, J. Saynisch, F. Zus, S. Heise, and J. Wickert. A Phase-Altimetric Simulator: Studying the Sensitivity of Earth-Reflected GNSS Signals to Ocean Topography. IEEE Trans. Geosci. Remote Sensing, 54(11):6791–6802, 2016.

Jens Wickert, Estel Cardellach, Manuel Martin-Neira, Jorge Bandeiras, Laurent Bertino, Ole Baltazar Andersen, Adriano Camps, Nuno Catarino, Bertrand Chapron, Fran Fabra, Nicolas Floury, Giuseppe Foti, Christine Gommenginger, Jason Hatton, Per Hoeg, Adrian Jaggi, Michael Kern, Tong Lee, Zhijin Li, Hyuk Park, Nazzareno Pierdicca, Gerhard Ressler, Antonio Rius, Josep Rosello, Jan Saynisch, Francois Soulat, C. K. Shum, Maximilian Semmling, Ana Sousa, Jiping Xie, and Cinzia Zuffada. GEROS-ISS: GNSS REflectometry, Radio Occultation, and Scatterometry Onboard the International Space Station. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens., 9(10, SI): 4552–4581, 2016.

L. Zhang, H. Dobslaw, T. Stacke, A. Güntner, R. Dill, and M. Thomas. Validation of terrestrial water storage variations as simulated by different global numerical models with grace satellite observations. Hydrol. Earth Syst. Sci., 21(2):821–837, 2017. 

• OMCT-Modell