Convolution and Deconvolution

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Presentation transcript:

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]; C = [0.2500 1.0000 0.6250 0 1.6250 -0.3750]

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.

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

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)

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)

Convolution

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)

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

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

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

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

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

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

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

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

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

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

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

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

Convolutional model

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

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

W E

W E

W E

W E

W E

W E

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

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

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

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

Very sharp break