Required smoothness constraints

After each time-step, we perform a further check on the interpolated temperature anomalies to ensure that they satisfy certain vertical and temporal smoothness conditions in their basin-averaged temperature anomalies. These checks are performed individually for the Atlantic, Pacific, Indian, and Southern Ocean basins (the Arctic is not considered for this check due to data sparsity) using information about the natural rates of change derived from observations during the Argo Era (defined as 2005 onwards). These checks are only performed for time-steps prior to 2005. The boundaries for these ocean basins are the same as defined in the main text and utilize the WOA 1-degree mask⁶³.

First, we calculate the change in basin-averaged temperature anomaly at each depth level from the current temporal iteration to the previous temporal iteration, Δθt(basin,z), from the interpolated temperature anomalies during the well-sampled Argo period (2005–2019). We also calculate the difference in basin-averaged temperature anomaly at each iteration from one depth level to the previous depth level, Δθz(basin,z), for the Argo period. This yields 28 values of Δθtand Δθzfor each basin and depth level. We then require that Δθt(basin,z) and Δθz(basin,z) for the current iteration not exceed the 3rd standard deviation (σ) of Δθt(basin,z) and Δθz(basin,z) during the Argo period. That is,

If a network does not pass both time and depth constraints for all ocean basins, then that network is rejected and the network must start over at the same time-step and depth interval. To allow for some additional flexibility, in the event that a network is rejected more than five times in a row, the run that produced the lowest average exceedance of these smoothing constraints is accepted and the procedure continues as usual. This is an uncommon occurrence, appearing only a few times in a single run from 2019 to 1946, but it most often occurs due to sharp changes in the temperature anomalies of the deep Pacific during the early 1970s.