By Carlamaria Maderna, Paolo Maurizio Soardi

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0 Eog(Ap) versus l o g ( A t ) 0 log(Ap’) versus l o g ( A t ) where Ap’ = dAP =At-dAP dlog (At) dAt The advantage of using the log-log plot is that it is able to display the whole data and show many distinct characteristics in a single graph. The pressure derivative plot with the pressure plot has two main advantages over the pressure plot alone from two different aspects: one is model recognition and the CHAPTER 3. CONFIDENCE INTERVALS 25 other is parameter estimation. First, from the aspect of model recognition, the pressure derivative plot reveals more characteristics of the response than the pressure plot.

Y,) for a fixed value of 6 . Given that Y = y (x= yl,. . ,Y, = yn) is observed, f(yl6) may be regarded as a function not of y but of 6 . The function of 6 defined by: is called the likelihood function. It should be mentioned that in the probability density function f ( y l 6 ) y and 9 are considered as random and as fixed variables, respectively, while in the likelihood function L(6ly) y is considered to be observed values and 8 is considered to be varying over all possible parameter values. Even if 6 can vary over all possible parameters in L(6ly),6 is not regarded as random.

When /3 = 1, the distribution becomes the double exponential distribution. When p tends to -I, the distribution tends to the uniform (rectangular) distribution. 27) However, the form of the error distribution is generally unknown in inverse problems. 2, the normal distribution assumption is the most unbiased assumption due to the Central Limit Theorem. This is the statistical reason why the least squares method is generally employed. 4 Bayesian Inference In this section, some basic principles of Bayesian inference are described.