We study the decomposition and compression of one-way wave propagation and imaging operation using wavelet transform. We show that the matrix representation of the Kirchhoff imaging operator (Kirchhoff migration operator) in space domain is a dense matrix, while the compressed beamlet- operator matrix which is the wavelet decomposition in the Kirchhoff operator, is a highly sparse matrix. The beamlet imaging operator represents the backpropagation of multiscale orthonormal beams (beamlets) at different positions with different angles. The beamlet-operator behaves differently in different wavelet bases. For sharp and short bases, such as the Daubechies 4, both the interscale and intrascale coupling are strong. On the other hand, the interscale coupling is relatively weak for smooth bases, such as higher-order Daubechies wavelets, Coiflets, and spline wavelets. The images obtained by the compressed beamlet operators are almost identical to the images from a full-aperture Kirchhoff operator. Compared with the conventional limited-aperture Kirchhoff migration (imaging), beamlet migration (imaging) can retain the wide effective aperture of a full-aperture operator, and hence achieves higher resolution and image quality with reduced computational cost. The compression ratio of the imaging operator ranges from a few times to a few hundred times, depending on the frequency, step length and the wavelet basis.