Adaptive curvelet-domain primary-multiple separation

The forward model

Without the wavelet and source directivity, the predicted multiples can be regarded as some scaled (along the time and receiver or offset axes) version of the true multiples. Mathematically, this nonstationary 'scaling' can be represented by a pseudo-differential operator. For our application, this operator acts on shot records or on common-offset panels and applies a location, frequency and dip-dependent zero-phase scaling. By applying a matrix-vector multiplication to the predicted multiples, this operator models the true multiples in the data, i.e.,

$$\mathbf{s}_2=\tensor{B}{\breve{\mathbf{s}}}_2,$$ (1)

where $\tensor{B}$ is a full positive-definite matrix, implementing the action of the pseudo-differential operator and ${\breve{\mathbf{s}}}_2$ represents the predicted multiples, calculated with single-windowed convolutional matched filtering. Relating the predicted multiples to the true multiples offers flexibility to model amplitude mismatches. Note, however, that this model cannot incorporate kinematic shifts, since pseudo-differential operators are unable to move wavefronts.

By compensating for the source wavelet and directivity, via a conventional local matched-filtering procedure, the pseudo-differential operator becomes zero order and permits a diagonal curvelet-domain decomposition (Herrmann et al., 2007c),

$$\mathbf{s}_2\approx\tensor{C}^T \{\mathbf{w}\}\tensor{C}{\breve{\mathbf{s}}}_2,\quad \{{w}\}_{\mu\in{\mathcal{M}}}>0$$ (2)

with $\tensor{C}$ the 2D discrete curvelet transform (see e.g.  Candes et al., 2006; Hennenfent and Herrmann, 2006) and $\mathbf{w}$ the curvelet-domain scaling vector and ${\mathcal{M}}$ the index set of curvelet coefficients. Since we are using the curvelet transform based on wrapping, which is a tight frame, we have $\tensor{C}^T\tensor{C}=$$\mbox{$\tensor{I}\,$}$ and the transpose, denoted by the symbol $\,^T$ , equals the pseudo inverse.

In this approximate forward model, for which precise theoretical error estimates exist (Herrmann et al., 2007c), the predicted multiples are linked to the actual multiples by a simple curvelet-domain scaling. This curvelet-domain scaling applies a location, scale and dip dependent amplitude correction. Since the matrix $\tensor{B}$ is positive-definite, the entries in the scaling vector, $\mathbf{w}$ , are positive. This approximate formulation of the forward model forms the basis for our curvelet-domain matched filter.

Adaptive curvelet-domain primary-multiple separation