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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 *l*^{1} norm. All these methods assign long-tailed density
functions to the model parameters. Figure 1 shows an exponential
(related to the *l*^{1} norm) and a Gaussian distribution (related to the *l*^{2} 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.

**distri
**

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

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Stanford Exploration Project

4/27/2000