Tomographic inversion using L1-regularization of Wavelet Coefficients

Like most geophysical inverse problems, the inverse problem in seismic tomography is underdetermined, or at best offers a mix of over- and underdetermined parameters. One usually regularizes the inverse problem by minimizing the norm (|m|) or roughness of the model (|∇ m| or |∇2 m|) to obtain a solution that is void of unwarranted structural detail. The notion that we seek the 'simplest' model that is in agreement with a given data set is intuitively equivalent to the notion that the model should be describable with a small number of parameters. But clearly, limiting the model to a few Fourier coefficients, or large scale blocks, does not necessarily lead to a geophysically plausible solution. We investigate if a wavelet basis can serve as a basis with enough flexibility to represent the class of models we seek. We propose a regularization method based on the assumption that the model m is sparse in a wavelet basis, meaning that it can be faithfully represented by a small number of nonzero wavelet coefficients. This allows for models that vary smoothly without sacrificing the sharp boundaries by a smoothing operator to regularize the inversion. To regularize the inversion, we minimize I= ∥ d - Am ∥2 + 2 τ ∥ w ∥1, where w is a vector of wavelet coefficients (m=Ww), τ the damping parameter, d-Am the vector of data residuals and 1 and 2 denote the ℓ1 and ℓ2 norm, respectively. the system is solved using Landweber iteration: w(n+1)= Sτ [ WATd + (I - WATAWT)w(n)], where Sτ is a soft thresholding operator (Sτ(x)=0 for |x|<τ and x ± τ elsewhere). In synthetic tests on a 2D tomographic model we show that minimizing the ℓ1 norm of a wavelet decomposition of the model leads to tomographic images that are parsimonious in the sense that they represent both smooth and sharp features well without introducing significant blurring or artifacts. The ℓ1 norm performs significantly better than an ℓ2 regularization on either the model or its wavelet decomposition. In particular, ray-path associated artifacts are almost completely suppressed.