  In this paper we consider the averaging of local field-theoretic Poisson
brackets in the multi-dimensional case. As a result, we construct a local
Poisson bracket for the regular Whitham system in the multidimensional
situation. The procedure is based on the procedure of averaging of local
conservation laws and follows the Dubrovin - Novikov scheme of the bracket
averaging suggested in one-dimensional case. However, the features of the phase
space of modulated parameters in higher dimensions lead to a different natural
class of the averaged brackets in comparison with the one-dimensional
situation. Here we suggest a direct procedure of construction of the bracket
for the Whitham system for $d > 1$ and discuss the conditions of applicability
of the corresponding scheme. At the end, we discuss canonical forms of the
averaged Poisson bracket in the multidimensional case.
