GNGTS 2015 - Atti del 34° Convegno Nazionale

18 GNGTS 2015 S essione 3.1 context of a NN optimization, an excessively complex network generates overfitting. Generally, overfitting occurs when a model describes random error or noise instead of the underlying relationship. Amodel that has been overfitted will generally have poor predictive performance. In the specific case, the overfitting associated with the NN method is visible by comparing Figs. 2a and 2c. In these figures, we note that the RPT derived from the NN approach tends to reproduce the scatter visible in the RPT derived from the recorded logs. Such scattered trend is clearly related to residual noise contamination in the well-log data or inaccurate measurements and it has no physical meaning. As a clarifying demonstration of the overfitting problem, we perform a blind test in which the petrophysical relations expressed by the four RPMs are used to predict the elastic properties in a nearby well that was not used in the estimation process of the RPMs (Fig. 3e). This well was drilled in the same target area and through geological formations with similar characteristics. This test is also aimed at quantifying the prediction capability of each rock-physics model. For the lack of space, we show the results of the blind test obtained for the S-velocity only that is the more difficult parameter to predict as demonstrated in Fig. 1. In Fig. 3e we note that, thanks to its general validity, the TRPM approach gives the best fit with the actual data. Conversely, the empirical, data-driven, approaches show lower correlation coefficients than TRPM. In particular, the NN approach is characterized by the lowest correlation coefficient (Fig. 3f), thus confirming that the overfitting problem is often associated with a sub-optimal prediction capability. Moreover, the blind test allows us to discuss a fundamental difference between the empirical and the theoretical approaches, that explains the lower correlation coefficients that characterize all the empirical RPMs with respect to the theoretical one (Fig. 3f). Even if the input set of elastic and petrophysical properties, used in defining the rock-physics model, belong to wells drilled through geological formations with similar characteristics, the empirical, data-driven, approaches return slightly different models depending on the set of input data considered in the prediction procedure. This fact can be ascribed to errors and uncertainties that affects the measured elastic properties and to errors and approximations made in the formation evaluation analysis to derive the petrophysical properties. Differently, the TRPM result, being based on theoretical equations, is totally independent from errors and uncertainties in well-log measurements. Conclusions. We have analyzed the rock-physics models (RPMs) obtained by applying both theoretical and empirical approaches. ��� ���� ����� ������� ��� �������� ��� ��������� ������� The fair match between the measured and predicted elastic properties and between the actual and the predicted RPTs demonstrates the potential of all the considered methods to yield final equations that are capable of estimating the elastic properties from a set of input petrophysical properties. A very high correlation coefficients characterize the density estimates, whereas lower correlation coefficients characterize the predicted seismic velocities. This fact evidences that the relation that links density to the petrophysical parameters is simpler than the relations existing between the petrophysical parameters and Vp and Vs . In addition, the lower correlation coefficient observed for the Vs estimates might be due to the lower performance of the logging tools in measuring the S-velocity. We have shown that the non-linear GA and the linear SR methods return very similar equations, demonstrating that the relations linking the input petrophysical properties to the elastic attributes are, in this specific case, close to be linear. This fact makes the application of an empirical non-linear method useless for the case under examination. However, �� ���� in more complex geological settings the linear approach may not be enough to ensure a good match between measured and predicted properties. In these cases non-linear methods should be applied. Among the non-linear methods we tested, an important limitation of the NN over the GAmethod is the overfitting problem. ������� �������� �� ��� �� ������� ��� �������� ����� Another drawback of the NN method, not analyzed here, is related to its local nature. In a NN optimization, the weights associated with each neuron are usually randomly initialized and are subsequently adjusted using a gradient-based strategy. This demonstrates the importance of a good initial model to prevent convergence towards a local minimum in the case of a complex multiminima error function. Conversely, �� ������ ��� GA method (or