Supplementary material to "A global Data Assimilation of Moisture Patterns from 21,000–0 BP (DAMP-21ka) using lake level proxy records"

S1 Explanation of data assimilation parameters and equationThe following two equations describe the data assimilation methodology at each timestep:S1) = loc * ( e , b ) c( e , e )+ , (S2) In equation S1, the matrices xb and xa describe vectors of the prior and posterior respectively.Each matrix includes an ensemble of values for each climate variable and geographic location.The innovation (yye) is the difference between the proxy observations and the prior estimate of the proxy value as calculated by the proxy system model.The innovation is spread to each location and variable according to the Kalman gain matrix (K) which is a square root ensemble Kalman filter (Whitaker & Hamill, 2002).As quantified by equation S2, K is a function of the covariance between the prior estimate (ye) with each climate variable at every grid cell (xb).This is the invaluable step of DA which updates the target climate variables according to the information contained by proxy values and which creates a uniform gridded dataset.Two additional factors are also included in K: a localization radius matrix (wloc) and positive diagonal error covariance matrix (R).As the geographic coordinates in the xb matrix become more distant from the proxy observations, wloc can decrease from 1 to 0 according to a predefined localization radius and is unitless.For the DAMP-21ka reanalysis, the wloc matrix weights the innovation according to a fifth-order polynomial so that the K is reduced with distance from a given proxy sites and locations further than the cut-off radius are not updated (Gaspari & Cohn, 1999; Tardif et al., 2019).R represents the uncertainty of the proxy observation in the squared units of y e .Specifically, a radius of 8,000 km and R of 302 (900) was selected based on validation experiments to test the sensitivity of each parameter (Table S3).Overall, reconstruct skill is relatively insensitive to these parameter choices, but lake status reconstruction skill is generally highest with small localization radius and R values due to coherent trends among nearby lakes throughout many regions.However, skill metrics calculated using the withheld pollen data suggest that larger values may be more appropriate.We selected the chosen parameters based on their high skill as measured by correlation with calibrated precipitation records both globally and in North America which is the one region with high densities of both lake status and pollen proxy records.