How to obtain a sparse model?

Some authors have proposed different solutions to address the sparseness of the model space. Thorson and Claerbout (1985) developed a stochastic inversion scheme that converges to a solution with minimum entropy. Sacchi and Ulrych (1995) apply a very similar method with more degrees of freedom to the choice of parameters. Nichols (1994) uses a regularization term with the l¹ norm. All these methods assign long-tailed density functions to the model parameters. Figure 1 shows an exponential (related to the l¹ norm) and a Gaussian distribution (related to the l² norm). The Gaussian distribution will tend to smooth the model space, spreading the energy, whereas the exponential distribution will tend to focus the energy on a few peaks, neglecting average values, and thus leading to a sparse model.

 

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Figure 1 Exponential (left) and Gaussian (right) distribution with zero mean. The exponential distribution has the longer tail.