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Convolution and Deconvolution

Convolution and Deconvolution. Convolution means several things:. IS multiplication of a polynomial series IS a mathematical process IS filtering. Convolution means several things:. IS multiplication of a polynomial series. A * B = C. E.g., A= 0.25 + 0.5 -0.25 0.75]; B = [1 2 -0.5];.

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Convolution and Deconvolution

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  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= 0.25 + 0.5 -0.25 0.75]; B = [1 2 -0.5]; C = [0.2500 1.0000 0.6250 0 1.6250 -0.3750]

  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. Convolution in theTIME domainis equivalent toMULTIPLICATION in theFREQUENCYdomain 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 time 1/2 z 1/2 -1/4 3/4 -1/4 3/4 Reflection Coefficients with depth (m)

  10. 0 0 0 -1/2 2 1 0 0 0 0 0 0 0 0 0 0 x x x x x = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 +

  11. 0 0 0 -1/2 2 -1 0 0 0 0 0 0 0 0 0 0 0 x x x x x = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 +

  12. 0 0 0 -1/2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 x x x x x x x = = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 +

  13. 0 0 0 -1/2 2 1 0 0 0 0 0 0 0 1/4 0 0 0 0 1/4 x x x x x x x x = = = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 +

  14. 0 0 0 1/2 1/2 0 0 0 0 1 0 0 0 -1/2 2 1 0 0 0 0 x x x x x x x x x = = = = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 +

  15. 0 0 0 -1/8 1 -1/4 0 0 0 0 5/8 x x x x x x x x x x = = = = = = = = = = 0 0 0 -1/2 2 1 0 0 0 0 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 +

  16. 0 0 0 0 -1/4 -1/2 3/4 0 0 0 0 x x x x x x x x x x = = = = = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 0 0 0 -1/2 2 1 0 0 0 0 +

  17. 0 0 0 1/8 1 1/2 0 0 0 1 5/8 x x x x x x x x = = = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 + 0 0 0 -1/2 2 1 0 0 0 0

  18. 0 0 0 -3/8 0 0 0 -3/8 x x x x x x x = = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 + 0 0 0 -1/2 2 1 0 0 0 0

  19. 0 0 0 0 0 0 0 x x x x x x = = = = = = 0 0 0 1/4 1/2 -1/4 3/4 0 0 0 + 0 0 0 -1 2 -1/2 0 0 0 0

  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 = 0.2500 1.0000 0.6250 0 1.6250 -0.3750 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 2Calama, Chile (181207_1) • Refractions (mainly) and reflections

  25. W E

  26. W E

  27. W E

  28. W E

  29. W E

  30. W E

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

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

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

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

  35. Very sharp break

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