VẬT lý địa CHẤN 13 deco

Outline • • • • • • Correlation Auto-correlation Filter Convolution Deconvolution Wiener filter Numerical evaluation of Cross-correlation n−τ φ (τ ) = ∑ x y i + τ xy i i =1 xi: (i=0 n) yi: (i= n) φxy(τ) : (-m < τ < +m) m = max displacement In Fourier domain: Cross-correlation = Multiplication of Amplitude spectrum and Subtraction of Phase spectrum Reynolds, 1997 Cross-correlation function Autocorrelation Cross-correlation of a Function with itself n −τ φ xx (τ ) = ∑ x i + τ x i =1 xi = (i=0 n) i φxx (τ) = (-m < τ < +m) m = max displacement Auto-correlation of two identical waveforms Normalization of correlation Auto-correlation Cross-correlation φ xx (τ ) φ xx;norm (τ ) = φ xx (0 ) φ xy ;norm (τ ) = φ xy (τ ) φ xx (0 )φ yy (0 ) Auto-correlation: multiples Autocorrelation functions contain reverberations a) A gradually decaying function indicative of short-period reverberation b) A function with separate side lobes indicative of long-period reverberations: multiples General Filter Filter Operator Input function Output function Numerical implementation of convolution m y k = ∑ g i f k −i i =0 k = m+n gi = (i=0 m) fj = (j= n) In Fourier domain: Convolution = Multiplication (of Amplitudes and Addition of Phasespectrum) Example of Convolution m yk = ∑ g i f k −i i =0 fk f0 f1 f2 f3 0 -1 0 ½ 0 -1 2-1 -1 2-1 ½ ½ 2-1 -1 ½ 2-1 -1 ½ ½ ½ gk ½ 2-1 2-1 -1 2 ½ ½ ½ 2-1 ½ ½ g gg gg gg g g gg gg gg g yk 12 021 10 12 -2 021 g102 gg021 gg102 ggg012 -2 -½ ggg012 gg021 gg10 -½ ¼ g0 Convolution model of the Earth wt ( equivalent Wavelet) gt = kδt ∗ st ∗ nt ∗ pt ∗ e t Impulse of source Near-surface zone of source Source effect gt = w t ∗ e t + Noise Reflectivity of the Earth Additional modifying Effects (absorption, wave conversion) + Noise * = + = Aim of Deconvolution Theoretical: Reconstruction of the Reflectivity function Practical: Shorting of the Signal Suppression of Noise Suppression of Multiples Deconvolution Reverse of Convolution xt = wt ∗ e t et = xt ∗ w-1t => Inverse Filtering Problem: w(t) is in general not known, i.e w -1 (t) Can not be determined directly Principle of Wiener filtering Principle of Wiener-Filters Input-Function ∗ (known) Filter = Output-Function (wanted) g0 g1 … ∗ (known) f0 g 0f = y0 f1 g1 f0 + g f1 = y … … => Solve system of equations Wiener filter • Spiking deconvolution: desired output is a spike • Predictive deconvolution: attempts to remove the effect of multiples After trace Common shot-gathers just balancing after demultiplexing Afterfor spiking deconvolution Corrected wavefront divergence Yilmaz, 1987 Deconvolved gather: note the prominent reflections Undeconvolved gather: reverberating wavetrains Undeconvolved gathers Deconvolved gathers Before After Deconvolution Before After Deconvolution [...]... filter • Spiking deconvolution: desired output is a spike • Predictive deconvolution: attempts to remove the effect of multiples After trace Common shot-gathers just balancing after demultiplexing Afterfor spiking deconvolution Corrected wavefront divergence Yilmaz, 1987 Deconvolved gather: note the prominent reflections Undeconvolved gather: reverberating wavetrains Undeconvolved gathers Deconvolved gathers... w t ∗ e t + Noise Reflectivity of the Earth Additional modifying Effects (absorption, wave conversion) + Noise * = + = Aim of Deconvolution Theoretical: Reconstruction of the Reflectivity function Practical: Shorting of the Signal Suppression of Noise Suppression of Multiples Deconvolution Reverse of Convolution xt = wt ∗ e t et = xt ∗ w-1t => Inverse Filtering Problem: w(t) is in general not known,... wavefront divergence Yilmaz, 1987 Deconvolved gather: note the prominent reflections Undeconvolved gather: reverberating wavetrains Undeconvolved gathers Deconvolved gathers Before After Deconvolution Before After Deconvolution ... f2 f3 0 -1 0 ½ 0 -1 2-1 -1 2-1 ½ ½ 2-1 -1 ½ 2-1 -1 ½ ½ ½ gk ½ 2-1 2-1 -1 2 ½ ½ ½ 2-1 ½ ½ g gg gg gg g g gg gg gg g yk 12 021 10 12 -2 021 g102 gg021 gg102 ggg012 -2 - ggg012 gg021 gg10 - ¼ g0... 1987 Deconvolved gather: note the prominent reflections Undeconvolved gather: reverberating wavetrains Undeconvolved gathers Deconvolved gathers Before After Deconvolution Before After Deconvolution... of Wiener-Filters Input-Function ∗ (known) Filter = Output-Function (wanted) g0 g1 … ∗ (known) f0 g 0f = y0 f1 g1 f0 + g f1 = y … … => Solve system of equations Wiener filter • Spiking deconvolution: