In this paper, we consider the problem of Bayesian sequential estimation on a set of time invariant parameters. At every time instant, a new observation through a linear model is obtained where the observations are distorted by spatially correlated noise with unknown covariance, whereas in time, the noise samples are independent and identically distributed. We derive the joint posterior of the parameters of interest and the covariance, and we propose several approximations to make the Bayesian estimation tractable. Then we propose a method for forming a pseudo posterior, which is suitable for settings where estimation over networks is applied. By computer simulations, we demonstrate that the Kullback-Leibler divergence between the pseudo posterior and a posterior obtained from a known covariance decreases as the acquisition of new observations continues. We also provide computer simulations that compare the proposed method with the least squares method.