GNGTS 2014 - Atti del 33° Convegno Nazionale

GNGTS 2014 S essione 3.1 121 it the most stable method. “Stable” means that if we change an entry in the transformation matrix slightly or add some noise to the dataset, the calculated model will not change largely. The conjugate gradient method constructs Radon transform results with iterations. In this algorithm the generalized cross validation (GCV) function can be used as the stopping criterion (Trad et al. , 2003). In both damped least squares and singular value decomposition methods the Lagrange multiplier β is used to balance the data misfit and the model misfit. In conjugate gradient method this balance is done by GCV function. Generally the Lagrange multiplier β is chosen according to the maximum curvature of the L-curve. This choice coincides more or less with the minimum in the GCV curve, except that this minimum point slightly overfits the data (Oldenburg and Li, 2005). By using the conjugate gradient we can get a more compressed butterfly structure in the Radon transform domain compared with the other two methods. After the forward and inverse parabolic Radon transform, the reconstructed t-x domain data have a lower signal to noise ratio and more vertical artifacts, but along the offset axis the attenuation of the amplitudes of the seismic event is lower compared with that presented in data reconstructed by the other two methods. Figs. 1a to 1d show this effect. Fig. 1a displays a synthetic seismogram with only one parabolic event in the time window [0.4 s 1.6 s]. This parabolic event can represent a multiple after NMO correction. Figs. 1b to 1d show the differences between the data in Fig. 1a and the reconstructed data using these standard different Radon transform methods. Both FDS-RT and TFDS-RT compress the butterfly structures by adding a model weight in the objective function of the inverse problem. The model weight is expressed as (1) Fig. 1 – a) Synthetic seismogram with only one parabolic event. This parabola represents a multiple after NMO correction. (b) to (h) show the differences between the data in (a) and the data derived from forward and inverse parabolic Radon transform applied to the data in (a) by using different Radon transform methods. (b) to (d) are sequentially related with standard parabolic Radon transform with (b) damped least squares method, c) singular value decomposition method, d) conjugate gradient method; (e) to (h) are sequentially related with sparse parabolic Radon transform methods: e) FDS-RT, f) TFDS-RT, g) ISS-RT with the standard parabolic Radon transform calculating the initial model, h) ISS-RT with FDS-RT calculating the initial model.