We have developed a P- and S-wave decomposition algorithm based on windowed Fourier transforms and a localized low-rank approximation with improved scalability and efficiency for anisotropic wavefields. The model and wavefield are divided into rectangular blocks that do not have to be geologically constrained; low-rank approximations and P- and S-decomposition are performed separately in each block. An overlap-add method reduces artifacts at block boundaries caused by Fourier transforms at wavefield truncations; limited communication is required between blocks. Localization allows a lower rank to be used than global low-rank approximations while maintaining the same quality of decomposition. The algorithm is scalable, making P- and S-decomposition possible in complicated 3D models. Tests with 2D and 3D synthetic data indicate good P- and S-decomposition results.