# How to create the initial ensemble samples for EnKF

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.