## 2 Ground Profile Characterization Based on a Probabilistic Approach
### Statistical Uncertainty  
It has previously been mentioned that the soil strength itself can be considered a statistical variable. The size of the sample is related to the statistical uncertainty of this variable. For each soil type and stress level, a set of values are available that are named \(N_{SPT,1}\)\(N_{SPT,N}\). However, the arithmetic mean of all these values does not necessarily have to match the average of the statistical parameter to be measured. There is a difference between the two that in literature is known as statistical uncertainty. If the N\({}_{SPT}\) variable at a depth is assumed to follow a normal distribution and the relative standard deviation is assumed to take a value of \(\upsilon\) = 0.15, this information could be used to correct the previously obtained arithmetic mean. [PERSON] et al. [10,11] demonstrate that assuming that the \(N_{SPT}\) is a statistic variable with a normal distribution, the distribution function of this same variable once the test has been performed n-times and values, \(N_{SPT,1}\)... \(N_{SPT,N}\), have been obtained is also a normal distribution function. \[f(N_{SPT})(N_{SPT,1}\ldots N_{SPT,N})=\phi_{\mu,\sigma}(N_{SPT}) \tag{3}\]  
where:  
\[\mu=\frac{N_{SPT,1}+\cdots+N_{SPT,N}}{N} \tag{4}\]  
\[\sigma=\upsilon\times\mu\times\sqrt{1+\frac{1}{N}} \tag{5}\]  
It is observed that, unsurprisingly, as the number of N experiments increases, the deviation is closer to that assumed for the statistical variable. For a sufficiently high number of tests, the arithmetic mean will match the wanted average. One way of considering statistical uncertainty is to calculate the average taking into account how far away each individual outputs is from it. The method is as follows:  
1. From the initial outputs of N\({}_{60}\), the arithmetic mean is obtained. It is named \(N_{SPT,1}\). 2. The standard deviation could also be obtained from the initial outputs. However, since the relative standard deviation is already known, the following expression is proposed:  
\[\sigma_{1}=\upsilon\mu_{1} \tag{6}\]  
Figure 2: Dispersion with SPT corrected by method uncertainty. Since each result, \(N_{SPT}\), i has provability defined by the function \(\phi_{\mu\downarrow 0\uparrow}\) (\(N_{SPT}\)), this value can be used to weigh the average in the following way \[\mu_{2}=\frac{\phi_{\mu\downarrow 0\uparrow}(N_{SPT,1})\times N_{SPT,1}+ \cdots+\phi_{\mu\downarrow 0\uparrow}(N_{SPT,N})\times N_{SPT,N}}{N}\times C.\] (7) where \(C\) is a constant, calculated in such a way that \[\frac{\phi_{\mu\downarrow 0\uparrow}(N_{SPT,1})+\cdots+\phi_{\mu\downarrow 0 \uparrow}(N_{SPT,N})}{N}\times C=1.\] (8)
4. The process can be repeated several times by setting a convergence criterion. However, it has been verified that for a sufficient number of results (N > 10) in the first iteration a value close to the final value is already reached. 5. This procedure can start from all field data, without deleting those whose value exceeds twice the standard deviation, as it was done in the previous section. This is because these values have an exceptionally low weight in the second iteration, which is equivalent to overriding them. So, this procedure can be used to account for both the statistical uncertainty and the uncertainty of the method. In addition to this correction, a second physical-sense correction is entered into the SPT values. Vertical strength variations that can exist within the same stratum are gradual. Abrupt variations are likely introduced by the measurement method. Therefore, the measured value at one point must be influenced by the measured value at the upper and lower levels. Thus, the mean at a certain depth is corrected with the expression. \[N_{correctal,i}=0.4\times N_{i}+0.2\times(N_{i-1}+N_{i+1})+0.1\times(N_{i-2}+N _{i+2}) \tag{9}\]  
where \(N_{i}\) is the average value at depth i.  
Figure 3: Dispersion with SPT corrected by method and statistical uncertainty. Applying these two corrections, a corrected average and deviation from the SPT test it is eventually obtained.