Reconstruction of GRACE Mass Change Time Series Using a Bayesian Framework

Abstract

Gravity Recovery and Climate Experiment and its Follow On (GRACE (-FO)) missions have resulted in a paradigm shift in understanding the temporal changes in the Earth's gravity field and its drivers. To provide continuous observations to the user community, missing monthly solutions within and between GRACE (-FO) missions (33 solutions) need to be imputed. Here, we modeled GRACE (-FO) data (196 solutions) between 04/2002-04/2021 to infer missing solutions and derive uncertainties in the existing and missing observations using Bayesian inference. First, we parametrized the GRACE (-FO) time series using an additive generative model comprising long-term variability (secular trend + interannual to decadal variations), annual, and semi-annual cycles. Informative priors for each component were used and Markov Chain Monte Carlo (MCMC) was applied to generate 2,000 samples for each component to quantify the posterior distributions. Second, we reconstructed the new data (229 solutions) by joining medians of posterior distributions of all components and adding back the residuals to secure the variability of the original data. Results show that the reconstructed solutions explain 99% of the variability of the original data at the basin scale and 78% at the one-degree grid scale. The results outperform other reconstructed data in terms of accuracy relative to land surface modeling. Our data-driven approach relies only on GRACE (-FO) observations and provides a total uncertainty over GRACE (-FO) data from the data-generation process perspective. Moreover, the predictive posterior distribution can be potentially used for "nowcasting" in GRACE (-FO) near-real-time applications (e.g., data assimilations), which minimize the current mission data latency (40-60 days).

Keywords: Bayesian inference; GRACE (‐FO); Geodesy; MCMC; Mass change; Hydrology.

Figure 1

Flow chart of the method used to impute the missing solutions within and between Gravity Recovery and Climate Experiment and its Follow On (GRACE (‐FO)) missions. First, the informative priors were derived from the GRACE (‐FO) for a single grid point/basin time series as the ranges of the intercepts, slopes, variability, amplitudes, and frequencies of the annual and semiannual cycles in the GRACE (‐FO) time series, assuming an additive generative model describing the geophysical signal in GRACE data as long‐term variability (secular trend + interannual to decadal variability), annual, and semi‐annual. Second, we combined the likelihood data and the priors in the Markov Chain Monte Carlo sampling to generate posterior distributions for each of the component storages. Third, we merged the median of the posteriors for the component storages to reconstruct the full GRACE (‐FO) total water storage and its uncertainty at 95% credible interval. We added the residuals back to the observed time series to preserve the same variability as the original GRACE time series. We applied 5‐fold cross validations to validate the model internally and generate a predictive posterior distribution that can be used to infer the present and near future of the total signal.

Figure 2

Mass change in three spheres, [a] land (excluding Greenland Ice Sheet (GIS) and Antarctic ice sheet (AIS)), [b] Ocean, expressed as equivalent sea level in mm, and Ice sheets ([c] Greenland and [d] Antarctica). The black line is the sum of the median posterior distributions for long‐term variability and annual and semi‐annual signals. Residuals were added back to periods with existing Gravity Recovery and Climate Experiment observations. The resorted solutions are represented by the reconstructed signal only without accounting for the residuals. Credibility of estimates is outlined with two probability levels (66%; likely and 95%; extremely likely).

Figure 3

Modeled total water storage during the Gravity Recovery and Climate Experiment (GRACE) and GRACE‐Follow On gap as the sum of the median posterior distribution of long‐term variability (variability ≥12‐month, including secular trend and interannual‐decadal variations), annual and semi‐annual signals between April 2002 and April 2021, sampled from 2,000 steps using Markov Chain Monte Carlo with the No‐U‐Turn Sampling method. Uncertainties associated with these signals are provided in supplementary materials at 5% and 95% levels (Figures S2, S3 in Supporting Information S1).

Figure 4

[a] Markov Chain Monte Carlo regression model diagnostic test with coefficient of determination (r²). [b] Empirical cumulative density function (ecdf) for r² showing ≥80% of the grid cells have r ² ≥ 58%. Variabilities in the predictable signal and the residuals are provided in SI (Figure S5 in Supporting Information S1).

Figure 5

Four evaluation tests of Gravity Recovery and Climate Experiment (GRACE) and GRACE‐Follow On mass change reconstructed data over land with catchment land surface model (CLSM)‐total water storage between the April 2002 and April 2021 and the associated ecdf. [a] correlation coefficient, [b] normalized mean square error, [c] Nash–Sutcliffe Efficiency, and [d] ratio of variability. Results are hachured over 11 land glaciers for consistency because the CLSM model does not simulate permanent snow or ice.

Figure 6

Time series for the five reconstructed data from four studies for the gap period between Gravity Recovery and Climate Experiment (GRACE) and GRACE‐Follow On missions. The original data from the two missions are shown in red circles. Results from this study are plotted in black lines with the 95% of credible interval (light orange).