Joint least-squares inversion of up- and down-going signal for ocean bottom data sets

Joint inversion of up/down-going P wave

Joint inversion of up- and down-going signals for ocean-bottom data can potentially be a better imaging technique than migrating either signal alone, because it combines information from both sets of signals. Figure 4 summarizes the processing scheme for our algorithm. Ocean bottom data are first separated into acoustic up- and down-going components above the seafloor. The decomposed signals are then inverted to yield one optimally combined reflectivity image. The objective function for such an inversion is:

$$0 \approx \left[ \begin{array}{c} \mathbf L_{\uparrow} \ \mathbf... ...n{array}{c} \mathbf d_{\uparrow} \ \mathbf d_{\downarrow} \end{array} \right],$$ (1)

where $\mathbf L_{\uparrow}$ and $\mathbf L_{\downarrow}$ are modeling operators that produce up-going data ( $\mathbf d_{\uparrow}$ ) and down-going data ( $\mathbf d_{\downarrow}$ ) from the model space ($\mathbf m$ ). The up- and down-going operators can be defined in many ways with varying levels of difficulty and practicality. We use the adjoint of the acoustic reverse time migration (RTM) operator to formulate $\mathbf L_{\uparrow}$ and $\mathbf L_{\downarrow}$ . Two modified computational grids are used to forward model the lowest order of up- and down-going signals, namely the primary and the receiver ghost. The formulation of the modeling and its adjoint (RTM) operator is summarized in Figure 2 and Figure 3.

forward
Figure 2. Forward modeling of (a) primary-only and (b) mirror-only data. The algorithm involves cross-correlating the source wavefield ($U_s$ ) with the reflectivity model ($m$ ) to generate the receiver wavefield ($U_r$ ). Reciprocity is used here where the data, in common-receiver domain, are injected at the source location while the source wavelet is injected at the receiver location. Cross-correlation is done only with grid points below the seabed.

reverse
Figure 3. RTM of (a) primary-only and (b) mirror-only data. The algorithm involves cross-correlating the source wave field ($U_s$ ) with the receiver wave field ($U_r$ ) to generate the reflectivity model ($m$ ). Cross-correlation is done only with grid points below the seabed.

Flowchart
Figure 4. Pressure (P) and vertical particle velocity (Z) data are converted into up- and down-going data. The up- and down-going data are then migrated separately using a modified grid shown in Figure 3. Inversion is performed with residuals in the up/down data domain.

In the modified computational grid as shown in Figure 2, the primary signal is obtained by the cross-correlation of the source wavefields with the reflectivity. For the down-going receiver ghost, the receiver nodes are placed at twice the water depth, which effectively represents a reflection off the sea surface.

Joint least-squares inversion of up- and down-going signal for ocean bottom data sets