In spite of its widely recognized importance, little progress has been achieved for the cumulus parameterization in last decades. Although the cloud-resolving model (CRM) is nowadays widely accepted as a useful tool to verify these parameterizations, it has so far failed to play a more active role of critically diagnosing the formulation for the latters, in spite of the fact that CRMs are capable of simulating three-dimensional convective systems reasonably. One of the problems is traced to the mass-flux formulation widely used for cumulus parameterizations, which decomposes a convective system into an ensemble of sign-definite up- and down-drafts. However, it is not straightforward to decompose an actual realization from a CRM experiment into an ensemble such sign-definite drafts, because such a decomposition is not unique mathematically due the lack of admissibility (a mathematically requirement posed on any orthogonal complete function sets). The wavelets have been proposed (Yano et al. 2001a, b, JAS) as an alternative method to diagnose outputs from CRM simulations in order to objectively characterize the convective system. In the present talk, we further propose the wavelets as a diagnostic tool in order to construct a convective representation (no longer called 'parameterization', because the convective elements are more explicitly considered in this approach) in a more logical manner through such objective diagnoses. Some preliminary analyses towards this goal will be presented, which include tests for the capacity (compressibility) for efficient representation of convective systems by wavelets, and the wavelet representation of the vertical heat and moisture fluxes.