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**Chapter 5**

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### Pairwise observation

### An observation of the square time versus offset square

### An illustration of a common midpoint (CMP)

### A CMP gather after the normal moveout (NMO) correction (Gaussian noise)

### A CMP gather after the normal moveout (NMO) correction (non-Gaussian noise, specifically exponential distribution)

### A CMP gather after the normal moveout (NMO) correction (uniform noise)

### A CMP gather after the normal moveout (NMO) correction (Gaussian noise)

### Third-order moments

### Bispectrum in the domain

### Quadratic phase coupling

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Pairwise observation
An observation of the square time versus offset square
An illustration of a common midpoint (CMP)
A CMP gather after the normal moveout (NMO) correction (Gaussian noise)
A CMP gather after the normal moveout (NMO) correction (non-Gaussian noise, specifically exponential distribution)
A CMP gather after the normal moveout (NMO) correction (uniform noise)
A CMP gather after the normal moveout (NMO) correction (Gaussian noise)
Third-order moments
Bispectrum in the domain
Quadratic phase coupling
Linear system/ nonlinear system
Energy concentrations
Third-order moment
Third-order moment (Cont'd)
Delay
Autocorrelation functions
Ghost identification
Crosscorrelation of seismic traces
Seismograms corresponding to wave propagation through a 2-D random medium
Crosscorrelation and bicoherence correlation
Crosscorrelation and bicoherence
Crosscorrelation and bicoherence correlation
Crosscorrelation, coherence correlation, bispectral correlation and bicoherence correlation
Seismic responses
Crosscorrelation of the two signals
Crossbicorrelation map
Crosscorrelation of the two signals
Crossbicorrelation map of the two signals
Coherence correlation of the two signals
Ricker wavelet
Autobicorrelation map
Crossbicorrelation map

Images - Chapter 5
Pairwise observation on (square time, square, offset) can constitute scatter diagrams. The relationship between the square time and the square offset is here approximated with straight lines.

Suppose we are given two random variables, X and Y, and wish to measure how good a linear prediction can be made of the value of, say, $Y$ based upon observing the value of X. At one extreme, if X and Y are independent, observing X tells us nothing about Y. At the other extreme, if, say, Y = a X + b, observing the value of X immediately tells us the value of $Y$. However, in many situations in petroleum seismology, two random variables are neither completely independent or linearly dependent. For instance, the crossplot of porosity and shear modulus for gas-saturated, pure-quartz sandstones, as illustrated in Chapter 2, suggests that the porosity and shear modulus are neither completely independent nor perfectly linearly dependent. Square time ($t^2$) versus offset square (x^{2}) relationship shows an even-less-clear linear trend, although the theory predicts a linear relationship between t^{2} and x^{2}. Given this state of affairs, it is necessary to quantify how much can be said about one random variable by observing another in addition to determining the coefficients a and b, which characterize the linear correlation between the two random variables, X and Y. The quantity called the "correlation coefficient" offers us such a measure.

Images - Chapter 5
(a) An observation of the square time versus offset square in a scatter diagram. The experimental points in this diagram have been used to estimate the best linear relationships between the square time and the offset square in the least-squares sense (l_{2} norm) and in the least-absolute-values sense (l_{1} norm). (b) Gaussian pdf for various values of the pair (a,b). The experimental values in 5.5a were used in these computations. (c) The exponential pdf for various values of the pair (a,b). The experimental values in 5.5a were used in these computations. (d) An observation of the square time versus the offset square in a scatter diagram. Notice that in the part (a) of the previous figure and also in (a) here, all the experimental points are inliers, whereas here we have a couple of outliers. Due to these outliers, the best linear relationship between the square time and offset square in the least-squares sense (l_{2} norm) and differs from the relationship between the square time and offset square in the least-absolute-values sense (l_{1} norm). The linear relationship based on the l_{2} norm is erroneous, whereas the one based based on the l_{1} norm is still quite close to the actual linear relationship. (e) The Gaussian pdf for various values of the pair (a,b). The experimental values in 5.5d were used in these computations. (c) The exponential pdf for various values of the pair (a,b). The experimental values in 5.5d were used in these computations.

Images - Chapter 5 An illustration of a common midpoint (CMP) gather before and after normal moveout (NMO) correction. Here the CMP consists of five signals corresponding to five source-receiver pairs: (S1, R1), (S2, R2), (S3, R3), (S4, R4), and (S5, R5). Each source-receiver pair represents a seismic experiment with a single source, say, S1 and a single receiver R1. Note that after NMO correction, the traveltime from the source position to the reflection point and from the reflection point to the receiver becomes identical for all five source-receiver pairs.The angle θ is the incident angle.

Images - Chapter 5 (a) A CMP gather after the normal moveout (NMO) correction. It contains 500 traces, but only 10 are shown, for the sake of the clarity of the picture. For timestep t, we have added to the data Gaussian noise with zero mean and 0.1 variance. We computed (b) the mean, (c) the variance, (d) the skewness, and (e) the kurtosis for each timestep, using all 500 traces. Notice that the skewness and kurtosis are almost null because the random variables at each timestep are Gaussian.

Images - Chapter 5 (a) A CMP gather after the normal moveout (NMO) correction. It contains 500 traces, but only 10 are shown, for the sake of the clarity of the picture. For timestep t, we have added to the data non-Gaussian noise, specifically exponential distribution with a 0.1 mean. We computed (b) the mean, (c) the variance, (d) the skewness, and (e) the kurtosis for each timestep, using all 500 traces. Notice a significant increase in the values of skewness and kurtosis compared to the case of Gaussian noise in the previous figure.

Images - Chapter 5 (a) A CMP gather after the normal moveout (NMO) correction. It contains 500 traces, but only 10 are shown, for the sake of the clarity of the picture. For timestep t, we have added to the data uniform noise with zero mean and 0.1 variance. We computed (b) the mean, (c) the variance, (d) the skewness, and (e) the kurtosis for each timestep, using all 500 traces. Notice that the skewness is almost null, despite the fact that our noise is non-Gaussian. The reason for the skewness being almost null is that the uniform distribution is symmetrical.

Images - Chapter 5 (a) A CMP gather after the normal moveout (NMO) correction. It contains 500 traces, but only 10 are shown, for the sake of the clarity of the picture. For timestep t, we have added to the data Gaussian noise with a zero mean. The linear AVA effects were also included in the computation of the CMP gather. We computed the mean, variance, skewness, and kurtosis for each timestep, using all 500 traces. These quantities are shown in blue curves in (b), (c), (d), and (e). Notice that the skewness and kurtosis are almost null in both cases, which confirms that the data remain Gaussian, despite the introduction of the linear AVA effects. We have also computed the mean, variance, skewness, and kurtosis for the case in which the data contained the nonlinear AVA effect. These quantities are shown in red curves in (b), (c), (d), and (e). Notice that the skewness and kurtosis have significant increased, which show that the nonlinear AVA has rendered non-Gaussian the data used to produce the cumulants in red curves.

Images - Chapter 5 (a) Symmetry regions of the third-order moments. We can divide the third-order cumulants into six symmetrical sectors, I through VI. The knowledge of third-order cumulants in any of the six sectors allows us to reconstruct the entire third-order cumulant sequence. (b) Symmetry regions of the bispectrum. Knowing the bispectrum in the shaded triangular region is enough for a complete description of the bispectrum based on the symmetries described in Table 5.3. We assume that in our bispectrum plot Δt =1 s, as we did throughout this section. To interpret this plot for other sampling intervals, we simply have to change π in this plot by π/Δt.

Images - Chapter 5
The bispectrum in the domain. The bispectrum is defined in the triangular region denote IT (inner triangle) if the signal is stationary [i.e., it is null outside the IT region, C_{3}^{(X)}(ω_{1}, ω_{2}) = 0, outside the triangular region]. If the signal is nonstationary, then the bispectrum is nonnull in the triangular region, denoted OT (outer triangle). The total domain occupied by IT and OT is known as the principal domain of the bispectrum. We assume that in this plot Δt =1 s, as we did throughout this section. To interpret this plot for other sampling intervals, we simply have to change π in this plot by π/Δt.

Images - Chapter 5
Quadratic phase coupling, (a) power spectra and (b) amplitude spectrum of bispectra.

Our goal in this example is to analyze outputs of nonlinear systems in the case in which the input consists of cosine waves (see Chapter 4 for the definition and analysis of cosine waves). In particular, we will show that the power spectrum of the output of a nonlinear system can include frequencies that are not contained in the input signal. The phenomenon behind these apparent new frequencies in output signals of nonlinear systems is known as phase coupling. We will start by introducing this phenomenon, and then we will describe examples of nonlinear systems which can cause phase coupling.

As illustrated in this figure, we can detect the phase coupling in {Z(k)} by computing its bispectrum. Why is the phase coupling detected by the bispectrum and not by the power spectrum? The answer to this question is that, in the power spectrum, signals are treated as a superposition of statistically uncorrected cosine waves. Therefore the power spectrum is not able to detect interactions between frequencies, which give rise to phase coupling. This weakness of the power spectrum can be generalized to the outputs of all linear time-invariant systems when the inputs consist of cosine waves.