Regression against time-differenced GRACE data
In our analysis we choose to make use of GRACE and GRACE-FO mass time series and regress against summed climate indices, rather than differencing the data and regressing against raw indices. In the absence of noise and data gaps these would produce identical results once the results of the latter approach were cumulatively summed. However, since GRACE and GRACE-FO do contain high frequency noise and uneven sampling, using the former approach avoids substantial issues with the latter, namely that time-differencing the data amplifies high-frequency noise relative to low-frequency signal, resulting in the need to filter the differenced data. To illustrate that either approach is possible and broadly equivalent using real, noisy data, we repeated the regression for WAIS but according to Equation 3:
\(\text{dM}\left(t_{i}\right)=b+\sum_{k=1}^{3}{\left(c_{k}^{s}\sin{\left(2\pi f_{k}t_{i}\right)+}c_{k}^{c}\cos\left(2\pi f_{k}t_{i}\right)\right)+d\text{SAM}+e{Nino3.4}}\)(3)
Where dM is the time series of GRACE and GRACE-FO after interpolating to exactly monthly sampling using a spline interpolator, filling gaps shorter than 3 months only, and then time differencing. To reduce high frequency noise, the series were also smoothed with a 43 month (~3.5y), centered running average. SAM and ENSO indices were normalized relative to the 1971-2000 reference period and smoothed in the same way as the data. After regression we cumulatively summed the terms to reproduce the time series of M . The results for WAIS are shown in Fig. S12. Fig. S12c shows the cumulatively summed SAM+ENSO model components estimated using Equation 3 along with those estimated using Equation 1, with their respective 1-sigma uncertainties. The two models overlap within uncertainties, confirming that our conclusions are not dependent on the underlying methodology summarized in Equation 1. The magnitude of mass loss associated with SAM and ENSO in the time-differenced solution is sensitive to the choice of smoothing window; using smoothing a smoothing window of 2.5y reduces the 2002-2021 WAIS mass loss associated with SAM and ENSO by 30% while a window of 4.5y increases it by 30%. These sensitivities are in addition to those associated with the choice of reference period, which are as described in our main solutions.
Lagged response to climate forcing
We examined the temporal cross correlation of the GRACE data and the two cumulatively summed climate indices. For the ENSO lag, we estimated and subtracted the first three terms of Equation 1 for the EAIS and APIS time series where high ENSO correlation was evident. We then cross-correlated the residual GRACE time series with the Niño3.4Σ timeseries and found a lag of 5 months for the APIS and 7 month for WAIS and we adopted a constant 6-month lag for all solutions. This WAIS lag is identical to that identified in Amundsen Sea ice shelf height timeseries by ref (27 ). We did not find a strong lagged correlation between GRACE and SAM and so adopted a zero lag for cumulatively summed SAM.
Partial Variance Explained by SAM and ENSO
We calculated GRACE time series partial variance explained (R2) by each, or both, of the SAMΣ and Niño3.4Σ terms, with R2 taken as the square of partial correlation R. We use partial correlation which, unlike conventional correlation, allows for the effect of a set of other confounding variables to be removed, namely the linear and periodic terms and the SAMΣ or Niño3.4Σ terms as appropriate. As our focus is multi-year variance, we applied a median smoother with width 7 months to the GRACE series before computing these values. Values of R2 are shown in Figure 2, Fig. S4 and S5, and for the gridded dataset in Fig. S6. The variance explained for the gridded time series is often smaller than for the basins or ice-sheet-wide results because of noise averaging over larger regions, although cancellation of SAMΣ and Niño3.4Σ signal over larger regions also occurs, as noted in the main text.
Trend uncertainties from multivariate and univariate solutions
GRACE time series for Antarctica contain low frequency noise(55 ) and this must be considered when computing the uncertainties of regression parameters. We found that compared to conventional linear regression our solution of Equation 1 explains most of the power at the lowest frequencies in the GRACE series. We tested the fit of auto-regressive 1 and Generalized Gauss Markov models(56 ) to the residuals and found both models fit the residuals well. Compared to a white noise only solution we found mean parameter uncertainties a factor of 1.6 larger when using either of these models. For the univariate solution (Equation 2), the residuals had highest power at lowest frequencies and so we adopted a power-law model, producing a white-noise scale factor of 2.9. Despite the larger scale factor, the uncertainties on the linear trends from the univariate solution (Fig. 4, and Fig. S8 and S9) are smaller than the multivariate solution due, at least partly, to the additional terms in Equation 1 compared to Equation 2, effectively trading off precision and accuracy.