GNGTS 2014 - Atti del 33° Convegno Nazionale

122 GNGTS 2014 S essione 3.1 where [W m ] ii is the i th diagonal entry of the diagonal model weight matrix W m , m i is the i th element of the model vector m. m is in frequency-curvature (f-q) domain for FDS-RT while for TFDS-RT it is in τ-q domain. This indicates that FDS-RT and TFDS-RT compress the butterfly structures in frequency domain and in time domain, respectively. As a consequence FDS-RT has a degraded vertical resolution while TFDS-RT needs a heavy computing time. As the weight is related with the model, we need to resolve nonlinear inverse problems to calculate these two sparse Radon transforms. In our code we use iteratively reweighted least squares to calculate the final model and in each iteration the conjugate gradient is used. ISS-RT also compresses the butterfly structures iteratively. It uses the initial model computed with any Radon transform method, then in each iteration it applies the shrinkage operator to refine the result. The tests done show that among the three sparse Radon transform methods, the FDS-RT has the drawback of giving τ-q domain data with the lowest signal to noise ratio and the lowest resolution, but it preserves at best the amplitudes of weak seismic events and causes the smallest amplitude attenuation along the offset axis for each seismic reflection in the reconstructed t-x domain data. This will make it the most suitable method for demultiple. On the contrary, TFDS- RT gives τ-q domain data with the highest signal to noise ratio and the highest resolution, but cannot preserve weak seismic events and causes very strong amplitude attenuation along the offset axis in the reconstructed t-x domain data. ISS-RT gives τ-q domain data with a moderate signal to noise ratio and a good resolution. At the same time it is able to reconstruct in t-x domain the weak seismic events but causes some amplitude attenuation along the offset axis even if this attenuation is lower than that of TFDS-RT. If we consider the demultiple processing as a multiple estimation process followed by a subtraction of the estimated multiples, Fig. 1 illustrates the demultiple ability of the different Radon transformmethods. The FDS-RT appears to be the most efficient in this context. Message Passing Interface. For accelerating the Radon transform algorithms proper filtering, use of the conjugate symmetric property of FFT, selective use of the zero padding before FFT and the exploitation of the characteristics of the circulant matrices should be considered (Strang, 1986). However, recent developments of parallel computing suggest to use personal computer clusters to help further speed-up the calculation of the Radon transform coefficients. We parallelized our ISS-RT code by exploiting two for loops in the algorithm. To do the parallelization we used the MPI package from Octave-Forge. This package allows us to connect Octave with Open MPI. Although the current MPI package (Version 1.2.0) cannot be used to send structures or complex numbers properly, by using cell arrays we circumvent this problem. In the parallel computing code the master node divides the input into pieces. At the beginning it sends each slave node one piece. When a slave node has finished calculating and has sent the results back to the master node, the latter will send a new piece to the former if there are still pieces of input left. Parallel computing is not always faster than direct calculation. For example in the ISS-RT algorithm one step that calculates a 2D averaging filter is needed. This step is not in the two for loops, so it is outside of the parallelized portion of the code. When we used the damped least squares method to calculate the initial model of ISS-RT, the 2D averaging filter occupied a large percent of the whole computing time. In this case the parallel computing was not faster than the direct calculation according to Amdahl’s law. When we used FDS-RT (which can be parallelized) to calculate the initial model of ISS-RT, the acceleration by using MPI on the whole computation time required by ISS-RT was evident. This was tested on a parallel computer system made by three Fujitsu Siemens Esprimo P5720 computers. With 6 processors we achieved a 3.17 times speedup when we set the iteration in the FDS-RT part as 200 and split the frequencies into 25 parts. We parallelized the ISS-RT code as an example, the parallel computing by using MPI can be applied to other Radon transform algorithms in the same manner.