Page 345 - Acquisition and Processing of Marine Seismic Data

336 6. DECONVOLUTION

TABLE 6.12 Autocorrelation of an Arbitrary f i Series (i ¼ 0, 1, 2, 3, 4, 5)

Output
f 1
f 4
f 2
f 3
f 5
f 0
2 2 2 2 2 2
f 0 f 1 f 2 f 3 f 4 f 5 a 0 ¼ f 0 + f 1 + f 2 + f 3 + f 4 + f 5
f 0 f 1 f 2 f 3 f 4 f 5 a 1 ¼ f 0 f 1 + f 1 f 2 + f 2 f 3 + f 3 f 4 + f 4 f 5
f 0 f 1 f 2 f 3 f 4 f 5 a 2 ¼ f 0 f 2 + f 1 f 3 + f 2 f 4 + f 3 f 5
f 0 f 1 f 2 f 3 f 4 f 5 a 3 ¼ f 0 f 3 + f 1 f 4 + f 2 f 5
f 0 f 1 f 2 f 3 f 4 f 5 a 4 ¼ f 0 f 4 + f 1 f 5
f 0 f 1 f 2 f 3 f 4 f 5 a 5 ¼ f 0 f 5
a 6 ¼ 0
f 0 f 1 f 2 f 3 f 4 f 5
a 7 ¼ 0
f 0 f 1 f 2 f 3 f 4 f 5
a 8 ¼ 0
f 0 f 1 f 2 f 3 f 4 f 5
TABLE 6.13 Cross-Correlation of Desired Output f(t + α) and f(t) for α ¼ 3

Output
f 5
f 4
f 3
f 0 f 1 f 2 f 3 f 4 f 5 c 0 ¼ f 0 f 3 + f 1 f 4 + f 2 f 5
f 0 f 1 f 2 f 3 f 4 f 5 c 1 ¼ f 0 f 4 + f 1 f 5
f 0 f 1 f 2 f 3 f 4 f 5 c 2 ¼ f 0 f 5
c 3 ¼ 0
f 0 f 1 f 2 f 3 f 4 f 5
c 4 ¼ 0
f 0 f 1 f 2 f 3 f 4 f 5
c 5 ¼ 0
f 0 f 1 f 2 f 3 f 4 f 5

TABLE 6.14 The Actual Deconvolution Output Obtained by Convolving h i Filter Coefficients With Input f i Series

Deconvolution output
f 2
f 5
f 0
f 1
f 3
f 4
h 5 h 4 h 3 h 2 h 1 h 0 d 0 ¼ h 0 f 0
h 5 h 4 h 3 h 2 h 1 h 0 d 1 ¼ h 1 f 0 + h 0 f 1
h 5 h 4 h 3 h 2 h 1 h 0 d 2 ¼ h 2 f 0 + h 1 f 1 + h 0 f 2
h 5 h 4 h 3 h 2 h 1 h 0 d 3 ¼ h 3 f 0 + h 2 f 1 + h 1 f 2 + h 0 f 3
h 5 h 4 h 3 h 2 h 1 h 0 d 4 ¼ h 4 f 0 + h 3 f 1 + h 2 f 2 + h 1 f 3 + h 0 f 4
h 5 h 4 h 3 h 2 h 1 h 0 d 5 ¼ h 5 f 0 + h 4 f 1 + h 3 f 2 + h 2 f 3 + h 1 f 4 + h 0 f 5
h 5 h 4 h 3 h 2 h 1 d 6 ¼ h 5 f 1 + h 4 f 2 + h 3 f 3 + h 2 f 4 + h 1 f 5
h 5 h 4 h 3 h 2 d 7 ¼ h 5 f 2 + h 4 f 3 + h 3 f 4 + h 2 f 5
h 5 h 4 h 3 d 8 ¼ h 5 f 3 + h 4 f 4 + h 3 f 5
h 5 h 4 d 9 ¼ h 5 f 4 + h 4 f 5
h 5 d 10 ¼ h 5 f 5