・ Text S1.
This section supplements the procedure to calculate C(T) described in the main text. First of all, for the central GNSS station and the 30 surrounding stations, the VTEC during the training period of 120 minutes from T-135 minutes to T-15 minutes is approximated by septic function as follows;
(S1)
in which VTEC(t) is the VTEC value at time t andypre (t ) is the fitted septic function, andε is the fitting error. Then, the VTEC during the prediction period is predicted by substituting t = T-15 to T into the obtained ypre (t ). Figures S2 and Movies S1 show the VTEC(t ) and ypre (t ) calculated with the central station of 0214 and satellite PRN26 for T = 5:20 on the day of Tohoku-Oki earthquake (same with Figure S1). In this case, according to I&U16, the training period is T-135 min ≤ t < T-15 min, the prediction period is T-15 min ≤ t ≤ T, and the VTEC(t ) - ypre (t ) of the prediction period is anomaly X(t ). The prediction functionypre (t ) shows a pronounced temporal variation, far from the natural extension of the VTEC of training period, which may be the result of overfitting the faint variation that appears during the training period (circled in the Figure S3a). As a result, X(t) seems not to be an indicator of whether VTEC is as predicted during the prediction period, but rather an indicator of small fluctuations during the training period. Then, the correlation is calculated between anomalies of the central GNSS station and its surrounding 30 stations. The correlations are averaged over 30 station pairs to be C(T) as expressed in Equation 1 in the main text. Figure S2-b shows C(T) calculated with the PRN26 satellite and 0214 central station. Movie S1 shows C(T) and temporal variation of VTEC(t ) and ypre (t ) for various T for the 0214 central station. We can see how ypre (t ), or X (t ), magnifies the faint VTEC change during the training intervals. The large C(T) fluctuation corresponding to the large X(t) variation means that the faint VTEC changes are coherent between 30 stations in this case. The C(T) in Figures S2, S3 and Movies S1, S2 show large oscillation with a period of about 15-20 minutes. This oscillation period is about half of a typical MSTID period, reflecting the fact that C(T) folds the negative side of the VTEC sinusoidal variation, thus halving the period. Contrary to I&Us’ claim that C(T) can extract VTEC variation of seismic origin, it is an indicator that amplifies small fluctuations synchronized between nearby stations, such as MSTID.