1. Convolution and Deconvolution

2. Convolution means several things:
IS multiplication of a polynomial series
IS a mathematical process
IS filtering

3. Convolution means several things:
IS multiplication of a polynomial series
A * B = C
E.g., A= ]; B = [ ];
C = [ ]

4. Convolutional Model for the Earth
output
input
Reflections in the earth are viewed as equivalent to a convolution process between the earth and the input seismic wavelet.

5. Convolutional Model for the Earth
output
input
SOURCE * Reflection Coefficient = DATA
(input) (earth) (output)
where * stands for convolution

6. Convolutional Model for the Earth
SOURCE * Reflection Coefficient = DATA
(input) (earth) (output)
where * stands for convolution
(MORE REALISTIC)
SOURCE * Reflection Coefficient + noise = DATA
(input) (earth) (output)
s(t) * e(t) n(t) = d(t)

7. s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase)
Convolution in the TIME domain is equivalent to MULTIPLICATION in the FREQUENCY domain
s(t) * e(t) n(t) = d(t)
FFT
FFT
FFT
s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase)
Inverse FFT
d(t)

8. Convolution

9. CONVOLUTION as a mathematical operator
signal
has 3 terms (j=3) -1 2
-1/2
earth
Reflection Coefficient
has 4 terms (k=4)
1/4
1/4
1/2
time z
1/2
-1/4
3/4
-1/4
3/4
Reflection Coefficients with depth (m)

10. -1/2 2 1
1/4
1/2
-1/4
3/4 x = +

11. -1/2 2 -1
1/4
1/2
-1/4
3/4 x = +

12. -1/2 2 1
1/4
1/2
-1/4
3/4 x = +

13. -1/2 2 1
1/4
1/4
1/2
-1/4
3/4 x = +

14. -1/2 2 1
1/2 1
1/4
1/2
-1/4
3/4 x = +

15. -1/8 1
-1/4
5/8 x =
1/4
1/2
-1/4
3/4
-1/2 2 1 +

16. -1/4
-1/2
3/4 x =
1/4
1/2
-1/4
3/4
-1/2 2 1 +

17. 1/8
1 1/2
1 5/8 x =
1/4
1/2
-1/4
3/4
-1/2 2 1 +

18. -3/8 x =
1/4
1/2
-1/4
3/4 +
-1/2 2 1

19. x =
1/4
1/2
-1/4
3/4 + -1 2
-1/2

20. MATLAB %convolution a = [0.25 0.5 -0.25 0.75]; b = [1 2 -0.5];
c = conv(a,b)
d = deconv(c,a)
c =
matlab

21. Spiking Deconvolution
In order to compress seismic signal in time and whiten the spectrum.
Advantages: shows embedded signal in noise
Disadvantages: heightens noise

22. Convolutional model

23. Steps in Spiking Deconvolution
Calculate autocorrelation function (ACF)
Estimate second crossing of ACF in s
Conduct inverse filtering using ACF

24. Significant arrivals in SP 2 Calama, Chile (181207_1)
Refractions (mainly) and reflections

25. W E

27. W E

28. W E

29. W E

30. W E

31. W E

32. Significant arrivals in SP 5 (191207_1)
Refractions (mainly)

34. Significant arrivals in SP 6 (181207_3)
Refractions (mainly)

36. Significant arrivals in SP 4A (171207_1)
Refractions (mainly)

38. Significant arrivals in SP 4B (191207_2)
Refractions (mainly)

40. Very sharp break