Deconvolution in the time domain

Deconvolution in the time domain can be implemented in terms of the following fitting goal for each (x,z) location:

$$\bf Dr=u,$$ (4)

where $\bf D$ is a convolution matrix whose columns are downshifted versions of the source wavefield $\bf d$.

The least-squares solution of this problem is

$$\bf r=(D^{'}D)^{-1}D^{'}u. \nonumber$$

where ${\bf D^{'}}$ is the adjoint of $\bf D$.A damped solution may be used to guarantee $\bf D^{'}D$ to be invertible as in  

$$\bf r=(D^{'}D+\varepsilon^{2})^{-1}D^{'}u \nonumber$$

where $\varepsilon$ is a small positive number. Equation (6) can be written in terms of the fitting goals

$$\begin{eqnarray} \bf 0 &\approx & \bf Dr-u \\ \nonumber \bf 0 &\approx & \bf \varepsilon I r,\end{eqnarray}$$ (5)

where $\bf I$ is the identity matrix. This approach can be computational efficient if the time window is not too large and we use a Conjugate Gradient as optimization engine. However, it has the disadvantage of relying on a linear inversion process that may or may not converge to the global minimum. A way to overcome this problem, obtaining an analytical solution, is to implement equation (6) in the Fourier domain, as we do in the next section.