DOI: 10.4211/hs.6f048c865eaa4cd58ef8bf4f3495670f
title: Total Water Storage Anomalies over East Africa Predicted by a GRACE-Based Bayesian Spatiotemporal Mixed Effects Model

description: This dataset contains total water storage anomalies (TWSAs) over East Africa predicted from observations from the Gravity Recovery and Climate Experiment (GRACE) mission using a Bayesian spatiotemporal mixed effects model.  The model was also used to estimate missing observations from the GRACE mission.  We obtained the RL05M.1 CRI Filtered Version 2 (Wiese et al., 2016, Watkins et al., 2015) monthly mass grids of the NASA JPL global mascon solution from the JPL’s GRACE TELLUS site.  TWSAs for 40 mascons covering East Africa were extracted from the dataset for May, 2002 through August, 2016.  This dataset did not contain data for several months due to missing observations in the global mascon solution dataset.  The missing values were predicted by the model.

The following Bayesian spatiotemporal mixed effects model was used to generate the modeled dataset.  The GRACE TWSA data can be represented by the spatiotemporal mixed effects model: Zt = Xtβ + Yt + εt; where Zt is a vector of TWSAs observations at time t, Xt is a matrix of fixed seasonal affects, β is  a vector of fixed covariate values for the seasonal affects, Yt is a vector of the true, underlying process, and εt is a vector of errors error terms.  

The true TWSAs at time t can be modeled by the autoregressive process: Yt = ΦYt-1 +ηt; where Φ defines the spatial-temporal structure of the GRACE TWSA and ηt is a vector of errors error terms.  However, estimating Φ is computationally difficult because of its high dimensionality.  

Empirical orthogonal function (EOF) analysis (Cressie & Wikle, 2011) can be used to identify the principal spatial structures in the GRACE TWSA data.  The dimensionality of the model is reduced by modeling the spatial structure using EOFs: Yt = Mut +ηt; ut = Ξut-1 + ζt; where M is a matrix of fixed, time-invariant basis functions defined as the first p empirical orthogonal functions (EOFs) of the data, ut is a vector representing a rank reduced process at time t, Ξ is a diagonal matrix defined as diag(ξ1.. ξp), representing the eigenvalues corresponding to the EOFs, and ζt is a vector of random errors error terms.  EOF analysis greatly reduces the computation burden of estimating the spatial-temporal structure of the GRACE TWSA.

A Bayesian approach is used to estimate the stochastic distributions for the model parameters ut , u0 , σ2ζ , β , and ξj.  Bayesian priors are chosen for each parameter and Monte Carlo Makov Chain methods are used to estimate the distribution parameters following the algorithm:
I.	Initialize the parameter values
II.	Gibs sampler draws from the posterior conditional for parameters ut , σ2ζ , u0, and β
III.	Slice sampler draws from the posterior conditional for the parameter ξj
IV.	Repeat II and III until the Markov chain converges to a stationary distribution

The calculations to implement the model are provided as part of the data archive.

The SI dataset contains the following fields:
•	ID: mascon ID assigned by NASA JPL
•	Year: year of the TWSA
•	Month: month of the TWSA
•	Day: day of the TWSA
•	TWSA_Obs: observed TWSA (NA if missing) in cm
•	TWSA_Mod: observed TWSA in cm
•	CI05: lower limit of the 90% credible interval for the modeled value in cm
•	CI95: upper limit of the 90% credible interval for the modeled value in cm

This dataset was created on April 28, 2017.

Cressie, N., & Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. Hoboken, New Jersey: John Wiley & Sons, Inc.
Watkins, M. M., Wiese, D. N., Yuan, D.-N., Boening, C., & Landerer, F. W. (2015). Improved methods for observing Earth’s time variable mass distribution with GRACE using spherical cap mascons: Improved Gravity Observations from GRACE. Journal of Geophysical Research: Solid Earth, 120(4), 2648–2671. https://doi.org/10.1002/2014JB011547
D. N. Wiese, D.-N. Yuan, C. Boening, F. W. Landerer, M. M. Watkins. 2016. JPL GRACE Mascon Ocean, Ice, and Hydrology Equivalent Water Height RL05M.1 CRI Filtered Version 2. Ver. 2. PO.DAAC, USA. Dataset accessed [2017-02-07] at http://dx.doi.org/10.5067/TEMSC-2LCR5.