TWS was resolved as daily changes in water mass on a grid of 0.25° × 0.25° nodes. Changes in the GPS-derived vertical and horizontal components were mapped to a surface mass load using Green’s functions for a spherical self-gravitating layered Earth model (4, 19). We minimized the objective function in Eq. 3 for the vertical and horizontal elastic responses to the daily water mass changes, with spatial and temporal regularization terms, using a positively bounded L1 norm inversion following (39)Embedded Image(3)where Gw and dw are the weighted Green function matrix and data vector, respectively, and other terms are defined below. Gwmt = dw together with Smt = 0, and Umt = Umt−1 can be expanded as followsEmbedded Image(4)where W is a diagonal weighting matrix associated with the formal daily uncertainty of the GPS position and G are Green’s functions (n × p matrix) relating p number of water mass patches loading a spherically layered elastic Earth (19) to surface deformation observed at n number GPS stations, with subscript v and u denoting vertical and horizontal components, respectively. Embedded Image is a vector of length n containing the vertical component (v) of the observed GPS displacement (d) on day (t), mt is the water mass that we solve for each day (t), and superscripts e and s denote east-west and north-south component, respectively. The solution was also regularized with temporal and spatial smoothing to suppress large and unrealistic variations of water mass between patches in space and time. λ and β are time-invariant coefficients that control the strength of spatial and temporal smoothing, respectively. S and U are the smoothing operators that estimate the current model’s (mt) spatial curvature and the difference from the previous model (mt−1), respectively, achieved with a central and backward finite-difference approximation, respectively. The choice of these smoothing factor values is discussed later below. The above problem was solved for using an outlier-resistant L1 iterative solver following the method of (39)Embedded Image(5)where M is the leftmost term in Eq. 4, MT is the transpose (which contains the weighted G and the regularization terms), and b is the rightmost term in Eq. 4 (which contains the weighted data vector, with the regularization terms appended below). R is a diagonal weighting matrix, with diagonal elements that are the absolute values of the reciprocals of the residualsEmbedded Image(6)Embedded Image(7)Therefore, R is a nonlinear function of mt, and the nonlinear system in Eq. 5 was solved for using an iteratively weighted least squares solver algorithm (19) that repeats the inversion for a new R once the model and residual vectors converge. The initial solution used to estimate R was taken from a positively bounded L2 norm inversion mt = (MTM)−1MTb.

The horizontal data were equally weighted in the inversion as the vertical data. Although the data uncertainty is smaller for the horizontals than for the verticals (on average 2 mm for horizontal and 5 mm for vertical at 1σ), meaning slightly higher weights in W, water mass changes are primarily sensitive to vertical motions. Because mass loading primarily causes vertical surface motion, the Green’s function values that relate the two are almost an order of magnitude larger, and therefore, water mass changes are more sensitive to vertical GPS changes than to the horizontal GPS changes. This explains the minor difference in the solution when including the horizontal in the inversion. However, we noted that the main effect of including the horizontal data is that it appears to improve the fidelity of the solution, concentrating water mass and giving a less smoothed result. This is not surprising, given that the horizontals are sensitive to local mass changes. Uncertainties of the TWS solution were obtained from 10,000 Monte Carlo simulations of the GPS data given the daily uncertainty (fig. S18).

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