Seismic Reservoir Modeling. Dario Grana

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Seismic Reservoir Modeling - Dario Grana


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P left-parenthesis upper Y less-than-or-equal-to y overbar right-parenthesis equals upper P left-parenthesis mu Subscript upper Y Baseline plus sigma Subscript upper Y Baseline upper X less-than-or-equal-to y overbar right-parenthesis period"/> For example, if Y has a Gaussian distribution with mean μY = 1 and variance sigma Subscript upper Y Superscript 2 Baseline equals 4, then we define a new random variable X = (Y − 1)/2. The probability P(Y ≤ 2) is then equal to the probability P(X ≤ 0.5) ≅ 0.69. Similarly, P(−3 ≤ Y ≤ 3) = P(−2 ≤ X ≤ 1) ≅ 0.82.

      In general, the probability that a Gaussian random variable X takes values in the interval (μXσX, μX + σX) is approximately 0.68; in the interval (μX − 2σX, μX + 2σX) is approximately 0.95; and in the interval (μX − 3σX, μX + 3σX) is approximately 0.99. Therefore, there is a high probability that the values of the Gaussian random variable script upper N left-parenthesis upper X semicolon mu Subscript upper X Baseline comma sigma Subscript upper X Superscript 2 Baseline right-parenthesis are in the interval of length 6σX centered around the mean μX, even though the tails of the distributions have non‐zero probability for values greater than μX + 3σX or less than μX − 3σX.

      Because a Gaussian PDF takes positive values in the entire real domain, when using Gaussian distributions for bounded random variables, such as volumetric fractions, or positive random variables, such as velocities, one should truncate the distribution to avoid non‐physical outcomes or apply a suitable transformation (Papoulis and Pillai 2002), such as the normal score or logit transformations of the original variables.

      

      1.4.4 Log‐Gaussian Distribution

      We then introduce the log‐Gaussian distribution (or log‐normal distribution) that is strictly related to the Gaussian distribution. Log‐Gaussian distributions are commonly used for positive random variables. For example, suppose that we are interested in the probability distribution of P‐wave velocity. To avoid positive values of the PDF for negative (hence non‐physical) P‐wave velocity values, we can take the logarithm of P‐wave velocity and assume a Gaussian distribution in the logarithmic domain. In this case, the distribution of P‐wave velocity is said to be log‐Gaussian.

Graph depicts log-Gaussian probability density function associated with the standard Gaussian distribution in Figure.

      (1.34)f Subscript upper Y Baseline left-parenthesis y right-parenthesis equals StartFraction 1 Over StartRoot 2 pi sigma Subscript normal upper X Superscript 2 Baseline EndRoot y EndFraction exp left-parenthesis minus one half StartFraction left-parenthesis log left-parenthesis y right-parenthesis minus mu Subscript normal upper X Baseline right-parenthesis squared Over sigma Subscript normal upper X Superscript 2 Baseline EndFraction right-parenthesis comma

      where μX and sigma Subscript normal upper X Superscript 2 are the mean and the variance of the random variable in the logarithmic domain.

      The mean μY and the variance sigma Subscript upper Y Superscript 2 of the log‐Gaussian distribution are related to the mean μX and variance sigma Subscript upper X Superscript 2 of the associated Gaussian distribution, according to the following transformations:

      (1.35)mu Subscript upper Y Baseline equals exp left-parenthesis mu Subscript upper X Baseline plus StartFraction sigma Subscript upper X Superscript 2 Baseline Over 2 EndFraction right-parenthesis comma

      (1.36)sigma Subscript upper Y Superscript 2 Baseline equals mu Subscript upper Y Superscript 2 Baseline left-bracket exp left-parenthesis sigma Subscript upper X Superscript 2 Baseline right-parenthesis minus 1 right-bracket equals exp left-parenthesis 2 mu Subscript upper X Baseline plus sigma Subscript upper X Superscript 2 Baseline right-parenthesis left-bracket exp left-parenthesis sigma Subscript upper X Superscript 2 Baseline right-parenthesis minus 1 right-bracket comma

      (1.37)mu Subscript upper X Baseline equals log left-parenthesis mu Subscript upper Y Baseline right-parenthesis minus StartFraction sigma Subscript upper X Superscript 2 Baseline Over 2 EndFraction equals log left-parenthesis StartFraction mu Subscript upper Y Superscript 2 Baseline Over StartRoot mu Subscript upper Y Superscript 2 Baseline plus sigma Subscript upper Y Superscript 2 Baseline EndRoot EndFraction right-parenthesis comma

      (1.38)sigma Subscript upper X Superscript 2 Baseline equals log left-parenthesis StartFraction sigma Subscript upper Y Superscript 2 Baseline Over mu Subscript upper Y Superscript 2 Baseline EndFraction plus 1 right-parenthesis period

      A log‐Gaussian distributed random variable takes only positive real values. Its distribution is unimodal but it is not symmetric since the PDF is skewed toward 0. The skewness s of a log‐Gaussian distribution is always positive and is given by:

      (1.39)s equals left-bracket exp left-parenthesis sigma Subscript upper X Superscript 2 Baseline right-parenthesis plus 2 right-bracket StartRoot exp left-parenthesis sigma Subscript upper X Superscript 2 Baseline right-parenthesis minus 1 EndRoot <hr><noindex><a href=Скачать книгу