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As we know, for the ensemble Kalman filter (EnKF), we need to create a set of samples in the beginning and then to run the predict and analysis step. But for now I have a question of how to create the ensemble samples.

For example, the initial state of some system is $x^0$,suppose we also know the true initial state $x^t$, then we can compute the covariance by $P^0=cov(x^0,x^t)$, and we can create an ensemble that follows the multivariate normal distribution,$N(x^0,P^0)$.

Q1:I am not sure if this implement is a Monte Carlo method?

Q2:It seems that using MCMC method or Gibbs sampling can also produce a certain ensemble to follow a certain distribution. But for the above case, is there any difference? Does one use the $N(x^0,P^0)$ to create the samples or use MCMC method to create samples following $N(x^0,P^0)$?

Q3: However, if we only know the $x^0$, and do not know the true state, or covariance $P^0$, how can we estimate the $P^0$, or create the initial ensemble samples for EnKF.

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