Total Water Storage Anomalies over East Africa Predicted by a GRACE-Based Bayesian Spatiotemporal Mixed Effects Model

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.

本数据集包含了东非地区总水量存储异常（TWSAs）的预测结果，该预测基于重力恢复与气候实验（GRACE）任务观测数据，并采用贝叶斯时空混合效应模型进行。此外，模型亦用于估计GRACE任务中的缺失观测值。本研究从NASA喷气推进实验室（JPL）全球mascon解算的RL05M.1 CRI过滤版2（Wiese et al., 2016, Watkins et al., 2015）月度质量网格中，提取了覆盖东非的40个mascons从2002年5月至2016年8月的TWSAs数据。由于全球mascon解算数据集中的观测值缺失，导致该数据集在数月间缺失数据。这些缺失值通过模型进行了预测。

用于生成模拟数据集的贝叶斯时空混合效应模型可表示为：Zt = Xtβ + Yt + εt；其中，Zt为时间t的TWSAs观测向量，Xt为固定季节影响的矩阵，β为季节影响的固定协变量值的向量，Yt为真实基础过程的向量，εt为误差项向量。

时间t的真实TWSAs可以通过自回归过程进行建模：Yt = ΦYt-1 +ηt；其中，Φ定义了GRACE TWSA的空间时间结构，ηt为误差项向量。然而，由于Φ的高维性，估计Φ的计算难度较大。

经验正交函数（EOF）分析（Cressie & Wikle, 2011）可用于识别GRACE TWSA数据中的主要空间结构。通过使用EOF对空间结构进行建模，降低了模型的维度：Yt = Mut +ηt；ut = Ξut-1 + ζt；其中，M为由数据的前p个EOF组成的固定、时间不变的基函数矩阵，ut代表时间t的秩减少过程向量，Ξ为对角矩阵，定义为diag(ξ1.. ξp)，表示对应于EOF的特征值，ζt为随机误差项向量。EOF分析显著降低了估计GRACE TWSA空间时间结构的计算负担。

采用贝叶斯方法估计模型参数ut、u0、σ2ζ、β和ξj的随机分布。为每个参数选择贝叶斯先验，并使用蒙特卡洛Makov链方法根据以下算法估计分布参数：I. 初始化参数值；II. 吉布斯抽样的参数ut、σ2ζ、u0和β的后验条件；III. 切片抽样参数ξj的后验条件；IV. 重复II和III，直到Makov链收敛到稳定分布。

实现模型的计算方法作为数据存档的一部分提供。

SI数据集包含以下字段：• ID：由NASA JPL分配的mascon ID；• Year：TWSA的年份；• Month：TWSA的月份；• Day：TWSA的日期；• TWSA_Obs：观测到的TWSA（如果缺失则为NA）以厘米为单位；• TWSA_Mod：观测到的TWSA以厘米为单位；• CI05：模拟值90%可信区间的下限以厘米为单位；• CI95：模拟值90%可信区间的上限以厘米为单位。

该数据集创建于2017年4月28日。

Cressie, N., & Wikle, C. K. (2011). Spatio-Temporal Data Statistics. 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.