The term, Bayesian statistics is referred to a class of statistical methods that provide a coherent framework for learning and problem solving under uncertainty conditions. The foundation of Bayesian statistics is based on the theory of probability, which is understood as a measure of the plausibility and uncertainty of a statement.
To implement Bayesian statistics, conditional probability is calculated using the Bayes’ theorem. In traditional statistics, which is not founded on Bayes’ theorem, the probability is only associated with random experiments results, while the Bayesian formulation allows to compute probabilities of all statements or propositions.
The advantage of Bayesian statistics in comparison to traditional statistics is that by using the Bayes’ theorem and estimating the probability density functions for the unknown parameters, the method of testing hypotheses or estimating their confidence regions can be readily tackled by these approaches. Therefore, Bayesian methods face rapid expansion and affect many application areas, where the uncertainty is inherent in many processes involved.
In recent years, Bayesian-based techniques have been used for merging various types of observations and model outputs. In our group, we develop new Bayesian frameworks to extract land hydrology (surface and sub-surface) and surface deformation (due to post glacial redound) from GRACE/GRACE-FO observations. We also apply the Bayesian technique for densifying geodetic networks. The main developed Bayesian frameworks in our group are:
1) Dynamic Model Data Averaging (DMDA), that is formulated to merge multi-model data with GRACE/GRACE-FO data, see Mehrnegar et al. (2020)
2) Markov Chain Monte Carlo-Data Assimilation (MCMC-DA), as an extension of DMDA, to recursively estimate components of the TWSC, while accounting for temporal dependencies between the storage compartments, see Mehrnegar et al. (2021)
3) Constrained Bayesian (ConBay), which is a Bayesian formulation to use GNSS and GRACE/GRACE-FO data for a simultaneous separation of the effect of the post glacier rebound and terrestrial water storage changes, see Forootan and Mehrnegar (2022).
4) Regionalising ionosphere models for positioning applications applies a Bayesian formulation to use available models as a priory fields and update them using GNSS network observations, see Farzaneh and Forootan (2020).
A novel MCMC-Data Assimilation is introduced in Mehrnegar et al. (2021) to integrate GRACE data with water storage outputs of the water balance model (W3RA). MCMC-DA down-scales GRACE to explore water storage changes at much finer resolution.
A synthetic example to demonstrate the ability of a Bayesian method for regionalizing ionosphere fields, see Farzaneh and Forootan (2020).
Related Publications:
Forootan, E., Mehrnegar, N. (2022), A hierarchical Constrained Bayesian (ConBay) approach to jointly estimate water storage and post-glacial rebound from GRACE(-FO) and GNSS data. All Earth, 34, doi:10.1080/27669645.2022.2097768
Mehrnegar, N., Jones, O., Singer, M.B., Schumacher, M., Bates, P., Forootan, E.(2020), Comparing global hydrological models and combining them with GRACE by Dynamic Model Data Averaging (DMDA). Advances in Water Resources, 138, doi:10.1016/j.advwatres.2020.103528
Mehrnegar, N., Jones, O., Singer, M.B., Schumacher, M., Jagdhuber, T., Scanlon, B.R., Rateb, A., Forootan, E. (2021), Exploring groundwater and soil water storage changes across the CONUS at 12.5 km resolution by a Bayesian integration of GRACE data into W3RA. Science of the Total Environment, 758, doi:10.1016/j.scitotenv.2020.143579
Farzaneh, S., Forootan, E. (2020), A least squares solution to regionalize VTEC estimates for positioning applications. MDPI Remote Sensing, 12 (21), doi.10.3390/rs12213545