Calculation of inverse filters

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Problem 9.18

Assuming that the signature of an air-gun array is a unit impulse and that the recorded wavelet after transmission through the earth is , find the inverse filter that will remove the earth filtering. How many terms should the filter include?

Solution

The inverse filter (see problem 9.7) is a filter that will restore the source impulse, i.e.,

or in the frequency domain where ,

Thus,


(9.18a)

where .

Since has the magnitude , the magnitude of for all values of , and we can expand equation (9.18a) using the binomial theorem [see equation (4.1b)] and Sheriff and Geldart, 1995, equation (15.43):


(9.18b)

We first find neglecting powers higher than :

We can verify the accuracy of by multiplying by . We have

We see that the inverse filter is exact as far as the term and terms for higher powers are small. The overall effect is to create a small tail whose energy is 0.00149 or 0.1%.

To determine how the accuracy depends on the number of terms used in , we observe the effect on the product as we successively drop high powers in . Dropping the term in yields the product and the energy of the tail is now 0.01063 or 1.1%. If we want accuracy of at least 1%, we must therefore retain the term.

If we also delete the term in [but not in ] the product becomes and the energy of the tail is 0.01850 or 1.8%. If we go one step further and drop the term in , we get and the energy of the tail is 0.13107 or 13.1%.

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