# Assessment of Concrete Strength Using the Combination of NDT--Review and Performance Analysis
## 6 Relationship between the Statistical Parameters of the Model \"Trade-Off Effect\"  
Based on the dataset of [40] composed of triplet of \(V\), \(R\), and \(f_{core}\), the trade-off effect is analyzed on the SonReb method. In order to analyze the model calibration quality, and to study the effect of calibration on the model parameters, the number of cores (NC) that are subject to calibration is varied between the minimum number of 3 cores and a maximum of 20 cores. This procedure is repeated 10,000 times for each number of cores in order to study the stability of the repetitive process. From each repetition, a model with its own statistical parameters is estimated. This set of parameters varies from one repetition to another. The relationship between models' parameters is called \"trade-off\". This effect has only been studied for the single NDT method [23]. The authors reported that the relation between the two parameters of all the models having the same form from the literature is linear. On a large experimental database [13], similar results are presented, as can be seen in Figure 5. This figure presents the results for both \(R\) and \(V\) measurements. The colours indicate the number of cores used for calibration. It can be seen that increasing NC allows one to decrease the variation range of the parameters \"a\" and \"b\".  
Regarding the SonReb method, this effect has not been studied. For this reason, in this paper, the same dataset from [48] that is composed of on-site measurement of 205 triplet \(V\), \(R\), and \(f_{c}\) from a reinforced concrete building was used.  
From this database, the SonReb method was tested by calibration. Figure 6 illustrates the trade-off between the statistical parameters \"a\", \"b\", and \"c\" of the multivariable model for the combination of rebound and ultrasonic pulse velocity. Furthermore, this trade-off is then analyzed for different NC with 10,000 repetitions for each given number.  
For a given NC, each repetition (randomly chosen) has its own model and its own parameters (a, b, and c), which are different from one simulation to another. It is noticed that all the parameters (ai, bi, and ci) of the repetitions (I = 1 to 10,000) for a given NC are on the same plane (ci = f(ai,bi)). Moreover, it is also noticed that even if NC is varied, the sets (ai, bi, and ci) always remain on the same plane. Dispersion of the sets is decreased by as much as the NC increases. Figure 6 confirms that, in an appropriate reference frame, the parameters a, b, and c of a multivariable model are related by a linear relationship.  
It is clearly shown in Figure 6 that the variation of the NC slightly influences the values of the parameters a, b, and c of the 10,000 repetitions of each NC. These values become closer to the parameters of the whole population model (N) as the NC increases. Moreover, the variation of the NC has a significant influence on the degree of dispersion (standard deviation) of the parameters a, b, and c of each NC.  
Figure 5: Trade-off between model parameters (for 300 iterations) with single methods for different numbers of NC [40]. (**a**) \(R\) test results, power model, b = f(LNa), (**b**) \(V\) test results, exponential model, b = f(LNa).