A note on the higher moments of the Euler characteristic of the excursion sets of random fields

In this paper, a useful and suitable form of the second moment of the Euler characteristic (EC) of excursion sets of a random field with respect to a special threshold is presented, which can be used in numerical evaluations of the variance of EC. We also present an extension to the higher moments of EC. Finally, a new method of simulating is presented and the results are compared with the three older well known methods of simulation based on the moment estimation and monte carlo idea.