Inversion of up and down going signal for ocean bottom data

Mirror Imaging modeling opeators

The first-order implementation of $\bold L_{\uparrow}$, denoted $\bold L^{\uparrow}_{1}$ performs wave equation modeling in a model space without the sea surface as shown in Figure 5 (a). Our up-going modeling operator is now approximated as

$$\bold L_{\uparrow} \approx \bold L^{\uparrow}_{1} .$$ (9)

When the sea surface is removed, waves always return to the receiver going upward. There are obvious limitations to $\bold L^{\uparrow}_{1}$. It can only model primaries and internal multiples. Higher order water reverberations are excluded. Figure 6 shows some events that are captured and excluded by $\bold L^{\uparrow}_1$.

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Figure 6. Left: The first-order up-going operator captures all the primaries and internal multiples. Right: It does not capture higher-order water reverberations and any pegleg multiples with a bounce at the sea surface. [NR]

On the other hand, we can use the mirror imaging modeling operator to get a first-order estimate of down-going signals, denoted $\bold L^{\downarrow}_{1}$. This operator performs wave equation modeling by raising the receivers to twice the water depth level as shown in Figure 5 (b). Our down-going modeling operator is now approximated as

$$\bold L_{\downarrow} \approx \bold L^{\downarrow}_{1}.$$ (10)

The limitation of $\bold L^{\downarrow}_{1}$ is that it can only image direct arrival and pegleg multiples having only one bounce from the sea surface. Higher-order water reverberations and pegleg multiples with two or more bounces from the sea surface are excluded. Figure 7 shows some events that are captured by $\bold L^{\downarrow}_1$. To do mirror imaging, we had to assume that the sea bottom is flat.

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Figure 7. Left: The first-order down-going operator captures the direct arrival and pegleg multiples with only one reflection from the sea surface. Right: It does not capture higher-order water reverberation and any pegleg multiples with two or more bounces at the sea surface. [NR]

Inversion of up and down going signal for ocean bottom data