## 2 Materials and Methods
### Criteria for the Development of Empirical Laws
#### 2.3.1 Limitations When Establishing Empirical Laws  
The correct development of an empirical law must involve a physical sense of the established functional dependence, indicating which parameter depends on the rest of the variables. The fact that a function \(h(g)\) can be expressed in the form \(g(h)\) does not indicate that the latter reflects the dependence that occurs in nature. For example, the relation of fluid viscosity to its temperature does not imply that temperature depends on viscosity. Similarly, with respect to a fluid flowing through the pores of a porous medium, the fact that the cation exchange capacity (CEC) of the grains depends on the medium permeability does not imply that permeability depends on the CEC. One of the empirical law limitations is the possibility of using different functions that reach a similar degree of deviation with respect to the empirical data. In this process, it should be considered that the use of goodness-of-fit indicators to evaluate a relationship to a set of empirical data (e.g., correlation coefficient, mean deviation, \(R^{2}\)) may have limitations. For example, the least squares criterion to determine the fit coefficients of a linearisable function to a set of data, in which the variable \(x\) involves both very high and very low values, can provide fit coefficients that constitute a function that greatly deviates from the data if there is a significant disparity in one of the lowest values. To show this effect, consider two series of 23 data points, \(y_{1}(x)=10\cdot\exp(-0.2\cdot x)\), and \(y_{2}(x)=0.1\cdot x^{2}\) with (\(x=1\), \(\ldots\), 23), and two other series \(y_{1}\)'(\(x\)) and \(y_{2}\)'(\(x\)) with the same values except for an anomalous point of the lowest \(y\) values, for example, \(y_{1}\)'(23) = 0.0001 (or another very close to zero) instead of 0.1 and \(y_{2}\)'(1) = 0.0001 instead of 0.1. Figure 1 shows that the fitting functions \(y_{1}\)'(\(x\)) and \(y_{2}\)'(\(x\)) obtained using least squares to the \(y_{1}\)'(\(x\)) and \(y_{2}\)'(\(x\)) series substantially differ from \(y_{1}(x)\) and \(y_{2}(x)\). The evaluation criteria for the goodness-of-fit relationship, which accounts for the number of measurements (e.g., the use of deviation \(\sigma_{n-1}\) versus the standard deviation), should not be ignored. The indicator used in error minimisation processes should also be considered for cases involving a single variable or multiple variables by solving the Jacobian matrix, which is formed by the partial derivatives of the results with respect to the variables. Linear convergence that involves the absolute value of the difference can thus lead to different results than those obtained using a curved convergence, which involves the conventional least squares criterion (i.e., the root of the squared differences). In a similar sense, the use of scales of a differentiated range (i.e., choice of axes) for parameters is not only a question of graphic representation but that the goodness of fit obtained using different criteria (e.g., regression coefficient \(R^{2}\)) can lead to inaccurate results. For example, the relation between [PERSON]'s modulus of the ground \(E_{\mathrm{S}}\) and dynamic modulus \(E_{\mathrm{D}}\) (i.e., obtained from seismic propagation velocities) presents strong dispersion, which is seemingly eliminated if \(E_{\mathrm{S}}/E_{\mathrm{D}}\) is plotted against \(E_{\mathrm{D}}\) in a logarithmic scale. Furthermore, there are other procedures that the authors consider to be inappropriate for determining empirical laws. For example, the result of multiplying or dividing an existing relationship by a factor does not strictly result in the establishment of a new law but rather a modification of the previous law. In other cases, it is used as an incorrect criterion that the new relationship is dimensionally consistent with a previous one (note that it is not a classical procedure of dimensional analysis).