Formulation of the problem

We want to solve the deconvolution problem:

 

y = a * x , (8)

y being the data, x the unknowns, a the convolution filter. It can be written in the matricial form A.x=y, with:

$$x=(x_1,\ldots,x_{n_x})^T\;,\mbox{\hspace{1.0cm}} y=(y_1,\ldots,y_{n_y})^T\;,\mbox{\hspace{1.0cm}}n_y = n_a + n_x -1\;,$$

 

$$A=\pmatrix{a_1 & & & & \cr a_2 &a_1& &0 & \cr \vdots&a_2&\... ..._{n_a}\cr} \mbox{\hspace{1.5cm} ($n_y\times n_x$\space matrix)}$$ (9)

I will consider two kinds of deconvolution:

• predictive deconvolution: (y_i) = (y_i-1) * f, where f is the unknown predictive filter;
• deconvolution by a known wavelet: y = w * r, w being this wavelet, r unknown ``reflection coefficients''.