Numerical simulations of elastic wave propagations are the critical components for seismic imaging and inversion. Finite-difference schemes yield high efficiency but fail to ensure the accuracy of the high wavenumber components. The pseudo-spectral algorithm is accurate up to the Nyquist frequency, and it is efficient because of the high level optimization of the fast Fourier transform (FFT) algorithm. The calculation of derivatives in elastic wave propagation could employ the success of FFT optimization. A conventional spectral method consists of three procedures: a forward real to complex (R2C) FFT, a multiplication of the derivative term, such as a for first derivatives, and a complex to real (C2R) inverse FFT. For any even number, the R2C FFT is actually utilizing a half-sized C2C FFT, and then a further spectral manipulation is employed to obtain the spectrum. We propose an efficient scheme to calculate the derivatives for elastic wave propagation in which we apply a forward C2C FFT, and then multiply it with a set of coefficients, and finally a C2C inverse FFT is applied to complete the calculations. With such scheme, a 30% of efficiency has been achieved. In the end, we demonstrate the accuracy and efficiency on a seismic imaging project.


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