The index $\lambda$ includes both displacements and scale j. Intuitively, the wavelet transform divides the histogram of differences based on scale and location where each coefficient represents the solution to the EMD subproblem. For a single wavelet, the mass to be moved is proportional to the volume of $|\psi_j(x)|$, i.e. to $2^{-jn/2}$ and the distance to travel is proportional to the amplitude of the wave, i.e. $2^{-j}$. The sum of all distances is an approximation of the EMD, as formally defined in the equation \ref{eq:emd_embedding}. This can be seen as similar to the way packages are shipped over large distances. The route is divided into several parts representing large and small distances. Packets from nearby locations are merged at the end of the short-distance route segment to travel together. Then another merger of packages from the whole country is done to ship together to the destination