Inverse filter to remove ghosting and recursive filtering

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Problem 9.19a

A source is located 8 m below the base of the LVL. Given that the velocity below the LVL is m/s, that m/s in the LVL, the densities are , , the sampling interval ms, and that the reflected signal is , find the original wavelet using the inverse filter [see the discussion of equation (9.17a)]

Background

The LVL is discussed in problem 4.16, inverse filters in problem 9.7, ghosting as a filter in problem 9.17.

Following the discussion in problem 9.17, we write

where is the original (unghosted) signal, is the ghosted signal, the filter is where is the reflection coefficient at the base of the LVL, and accounts for the delay of the ghost relative to .

The inverse filter is


(9.19a)

The original signal is given by

In the time domain this becomes


(9.19b)

This process is called recursive filtering or feedback filtering.

Solution

The reflection coefficient involved in ghosting at the base of the LVL by a wave approaching from below is given by equation (3.6a), the value being

The filter term is which now becomes . To get , we have for the two-way traveltime from the source to the base of the LVL ms if ms. Thus,

and the inverse filter of equation (9.19a) is

Since , we can expand using equation (4.1b) and get

Omitting inverse filter terms higher than , we get

We get by multiplication as follows

Assuming the original wavelet was , the third term is close but not exact, the next three terms are quite small, but the final three terms are sizable and will cause the highly undesirable result of creating the appearance of a fictitious event.

We can verify this answer by adding and as follows:

Problem 9.19b

Find the original wavelet by recursive filtering.

Solution

Equation (9.19b) gives

that is, each element of is equal to the corresponding element of plus 0.79 times the element of that ocurred two time units earlier. Since , , equation (9.17a) shows that must also be causal, that is, , . Thus,

This result seems to be better than that obtained in part (a) only because the calculation terminated at . However, if we apply the rule to and subsequent elements, we would see a small tail added.

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