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By Cosan Ayan (Ed.)

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We choose to accept or reject the hypothesis that the slopes are the same based on this likelihood. Note that a) even if the slopes are the same, in a small number of cases we may reject that hypothesis because the statistical noise is unusually distributed. b) If the slopes are not all the same then the calculated value of σ2 is larger than what it would be if the slopes were the same and there were only random noise. This systematic difference reduces the calculated likelihood from the F test but in a small number of cases we may conclude that the slopes are the same when they are actually different.

We’ll discuss both tests below. SPWLA 48th Annual Logging Symposium, June 3-6, 2007 Making these statements quantitative, call the difference in the linear and quadratic fit residuals R. The expectation value of R is σ2 if p(z) is linear and is larger than this by the expectation value of the quadratic term if p(z) is quadratic. In either case the expectation value of the sum of the quadratic fit residuals, S2, is (N-(r+2)) σ2. Here (r+2) is the number of independent coefficients in the least square fit.

15) 2 Here we use the notation zi to denote the average of the variable z in run i. This means that we can find the slope of the best fit line without calculating the shifts. A crossplot of the individual line slopes and their run midpoint depths along with the weighted average slope of Eq. 14 is shown in Fig. 5. 315x + 9700 15980 15960 19900 20000 20100 20200 20300 20400 20500 SSD (ft) Fig. 6 First step in finding the most likely shifts. Choose dataset 1 (pink squares) as the reference dataset, draw the blue line with the best fit slope b through the run midpoint (green dot), then shift the other datasets' midpoints (green dots) along the black arrows to the blue line.

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