Data interpolation and noise removal

Applying a mask in equation (6) eliminates the contribution of the empty traces in the model space, making them invisible to the inversion. Therefore, by simply remodeling a data panel from the estimated model ${\bf \hat{m}}$ after inversion without the mask, the missing traces are reconstructed. Then, the interpolated data vector ${\bf d_{int}}$ can be estimated as follows:

$${\bf d_{int}}={\bf d}+{\bf (I-M)L\hat{m}},$$ (9)

where I is the identity matrix. Now, for the noise removal, we simply (1) apply a mute ${\bf K}$ in the radon domain that isolates and preserves the signal, and (2) transform the muted panel in the data space as follows:  

$${\bf n_{est}}={\bf MLK\hat{m}},$$ (10)

where ${\bf n_{est}}$ is the estimated signal (specular reflections and impinging source). The estimated noise ${\bf n_{est}}$ (diffracted energy and ambient noise) is obtained by subtracting the estimated signal from the input data:  

$${\bf s_{est}}={\bf d-MLK\hat{m}}.$$ (11)

Note that the estimated noise and signal in equations (10) and (11) are for the non-interpolated data. To compute the estimated noise and signal for the interpolated data, M must be removed in equations (10) and (11) and d must be replaced by ${\bf d_{int}}$ in equation (11).