CMP - MakingCMPs...

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    Making CMP’s From chapter 16  “Elements of 3D Seismology” by Chris Liner 
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    Outline Convolution and Deconvolution Normal Moveout Dip Moveout Stacking
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    Outline Convolution and Deconvolution Normal Moveout Dip Moveout Stacking
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    Convolution means several things: IS multiplication of a polynomial series IS a mathematical process IS filtering
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    Convolution means several things: IS multiplication of a polynomial series E.g.,  A= 0.25 + 0.5 -0.25 0.75]; B = [1 2 -0.5]; 0 1 2 3 0.25 0.5 0.25 0.75 A z z z z = + - + 0 1 2 2 0.5 B z z z = + - A * B = C C = [0.2500    1.0000    0.6250         0    1.6250   -0.3750]
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    Convolutional Model for the Earth input output Reflections in the earth are viewed as equivalent to a convolution   process between the earth and the input seismic wavelet.
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    Convolutional Model for the Earth input output SOURCE *  Reflection Coefficient = DATA (input)          (earth)                  (output) where * stands for convolution
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    Convolutional Model for the Earth (MORE REALISTIC) SOURCE *  Reflection Coefficient = DATA (input)          (earth)                  (output) where * stands for convolution SOURCE *  Reflection Coefficient + noise  = DATA (input)               (earth)                               (output) s(t)     *          e(t)                   + n(t)     =     d(t)
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    Convolution  Convolution in the     TIME  TIME  domain    is equivalent to     MULTIPLICATION MULTIPLICATION  in   in the     FREQUENCY FREQUENCY     domain s(t)     *          e(t)                   + n(t)     =     d(t) s(f,phase)  x  e(f,phase)  + n(f,phase)         = d(f,phase) FFT FFT FFT Inverse FFT d(t)
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    CONVOLUTION as a mathematical operator  0 j j j k k k D s e - = = 2 -1/2 -1 z Reflection Coefficients with depth (m) 1/4 1/2 -1/4 3/4 1/4 1/2 -1/4 3/4 Reflection Coefficient signal signal has 3 terms (j=3) has 3 terms (j=3) earth earth has 4 terms (k=4) has 4 terms (k=4) time
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    0 0 0 1/4 1/2 -1/4 3/4 0 0 0 0 -1/2 2 1 0 0 0 0 x x x x x = = = = = 0 0 0 0 0 0 +
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    0 0 0 1/4 1/2 -1/4 3/4 0 0 0 0 0 -1/2 2 -1 0 0 0 0 x x x x x = = = = = = 0 0 0 0 0 0 0 +
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    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 x x x x x x x = = = = = = = 0 0 0 0 0 0 0 0 +
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    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 x x x x x x x x = = = = = = = = 0 0 0 1/4 0 0 0 0 1/4 +
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    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 x x x x x x x x x = = = = = = = = = 0 0 0 1/2 1/2 0 0 0 0 1 +
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    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 x x x x x x x x x x = = = = = = = = = = 0 0 0 -1/8 1 -1/4 0 0 0 0 5/8 +
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    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 x x x x x x x x x x = = = = = = = = = = 0 0 0 0 -1/4 -1/2 3/4 0 0 0 0 +
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    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 x x x x x x x x = = = = = = = = 0 0 0 1/8 1 1/2 0 0 0 1 5/8 +
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    0 0 0 1/4 1/2 -1/4 3/4 0 0 0 0 0 0 -1/2 2 1 0 0 0 x x x x x x x = = = = = = = 0 0 0 -3/8 0 0 0 -3/8 +
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    0 0 0 1/4 1/2 -1/4 3/4 0 0 0 0 0 0 -1 2 -1/2 0 0 x x x x x x = = = = = = 0 0 0 0 0 0 0 +
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CMP - MakingCMPs...

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