text: **Extended Data Fig. 3 | Simulations. a**, Theoretically computed*r*2 stratified 
  by minor allele frequency between true genotypes and genotypes with 
  randomly introduced errors (Methods), emulating imputation accuracy plots. 
  **b**, Admixture simulation where sources P1 and P2 split 250 or 500 generations 
  ago and a pulse admixture event gives rise to a target population PX 50 
  generations ago. We vary demographic history, error rates, and sample sizes 
  and simulate 20 replicates for each scenario (see Methods for simulation 
  details). Admixture proportions are computed using an*f*4-ratio statistic and the 
  Twigstats cutoff is set to twice the source split time and the rare variant cutoff 
  is 5%. We plot two standard errors around the mean.**c**, qpAdm simulation taken 
  from21,23, as well as an adapted version where all population split times and 
  the admixture date are divided by 5. The Twigstats cutoff time is chosen to be 
  1200 generations (top) and 600 generations (bottom). We simulate 10 replicates 
  and plot two standard errors around the mean.**d**, Simulation adapted from23 of 
  a stepping stone model with 9 populations organised on a 1-dimensional grid as 
  shown, where individuals are able to migrate between adjacent fields. We run a 
  rotational qpAdm to fit population 4 using other pairs of populations to the left 
  and right as sources. We run 50 replicates and set the p-value of models with 
  inferred proportions outside of [0,1] to 0. We then compute the proportion 
  where a given pair achieves the best p-value (top) and show the median p-value 
  across these replicates (bottom). In all simulations in**b, c, d**, we sample N = 20 
  haploid sequences per population, except for one simulation in**b**, where we 
  sample N = 100 sequences. 