Attenuation Parameters of Blasting Vibration by Fuzzy Nonlinear Regression Analysis

Chen Li， Shufeng Liang， Yongchao Wang， Long Li and Dianshu Liu

（School of Mechanics and Civil Engineering, China University of Mining and Technology,Beijing 100083, China）

Abstract: In order to reduce the influence of outliers on the parameter estimate of the attenuation formula for the blasting vibration velocity, a fuzzy nonlinear regression method of Sadov’s vibra?tion formula was proposed on the basis of the fuzziness of blasting engineering, and the algorithm was described in details as well. In accordance with an engineering case, the vibration attenuation formula was regressed by the fuzzy nonlinear regression method and the nonlinear least square method, respectively. The calculation results showed that the fuzzy nonlinear regression method is more suitable to the field test data. It differs from the nonlinear least square method because the weight of residual square in the objective function can be adjusted according to the membership of each data. And the deviation calculation of least square estimate of parameters in the nonlinear re?gression model verified the rationality of using the membership to assign the weight of residual square. The fuzzy nonlinear regression method provides a calculation basis for estimating Sadov’s vibration formula’s parameters more accurately.

Key words: blasting vibration；attenuation parameters；fuzzy nonlinear regression；membership

The prediction of blasting vibration intens?ity is essential during the design of various blast?ing projects. Currently, there are four main methods for the prediction: Sadov’s vibration for?mula[1?2], neural network[3?4], numerical simula?tion[5?6], and single?hole waveform superimposed calculation[7?9]. Because of its definite physical meaning and convenient application, Sadov’s for?mula is a most commonly used method in engin?eering.

The key to Sadov’s vibration formula lies in the accurate determination of its parameters so that it can guide the blasting construction. In previous studies, the formula’s parameters were generally estimated by the least square method whose objective function is to minimize the resid?ual sum of squares, which experienced the devel?opment from linear[10]to nonlinear[11]. The object?ive function of nonlinear least square method is expressed as

During testing, due to the instability of test?ing conditions, improper operation of instru?ments, and errors in observation and recording, it is inevitable to generate data with high discrete?ness, called outliers. However, the least square method needs to accommodate all data points,that leads to the influence of outliers on the re?sidual sum of squares is far higher than that of other data points, and the attenuation formula’s parameter estimate is sensitive to outliers. When performing linear regression on test data, “2σ” or“3σ” principle (σ is the residual standard devi?ation) can be used to distinguish and eliminate outliers with residual analysis[12]. But it is too dif?ficult to do that in nonlinear regression. Usually the data was selected to be removed or retained by artificial judging according to the fitting res?ults. This step is subjective and easy to omit or delete the real test information, so it is im?possible to estimate the parameters accurately.

To solve the problem above, considering the fuzziness of blasting engineering[13], this paper puts forward a fuzzy nonlinear regression meth?od of Sadov’s vibration formula and verifies its rationality, aiming to obtain the optimal estima?tion of parameters in Sadov’s formula on the basis of the test data.

1 Fuzzy Nonlinear Regression Method

Sadov’s vibration formula is recommended to describe the attenuation of peak particle velo?city in China’s safety regulations for blasting(GB 6722–2014)[14], as shown as

It can be seen that solving Eq.(11) is an it?erative process of stepwise approximation, and the calculation flowchart in Fig.1 can represent its specific analysis procedure.

Fig. 1 Flow chart of calculation

The difference between the fuzzy nonlinear regression method and the nonlinear least square method is that the weight of residual square in the objective function can be adjusted according to the membership of each data, so that the high?er the membership of the data, the greater the impact on the regression result. When the fuzzi?ness of blasting engineering and the influence of outliers are not considered, the membership of all data is 1, that is, uA?[Vi]=1 for any i. Substitut?ing uA?[Vi] into Eq.(11), Eq.(11) will be trans?formed into the equation of nonlinear least square method in 11 references. In this case, the nonlinear least square method can be regarded as a particular case of the fuzzy nonlinear regres?sion method. Therefore, the fuzzy nonlinear re?gression method takes the membership of the test value relative to its estimate value as the resid?ual weight is more conforming to the actual situ?ation.

2 Analysis of Engineering Case

Taking the blasting vibration test data of an open?pit mine as an example, the nonlinear least square method and the fuzzy nonlinear regres?sion method estimated Sadov’s vibration formula’s parameters, respectively. Field test data are shown in Tab.1.

Tab. 1 Blasting vibration test data

Given the accuracy eK=eα=0.000 05, iterat?ive calculations were performed according to the analysis procedure in Fig.1. And the calculation results are shown in Tab.2 and Fig.2.

It can be seen from Fig.2 that, compared with the nonlinear least square method, the fuzzy nonlinear regression curve fitted most data points better. The membership of each data point in the last iteration process was extracted and plotted,as shown in Fig.3. It can be seen intuitively that,except for the seventh data, the memberships of all other data were above 0.5 and close to 1 in the 20 sets of test data. It showed that the fuzzy nonlinear regression method only makes small ad?justments to the majority of the residual weights.It mainly reduces the residual weight of the sev?enth data and its impact on the fitting results.

Tab. 2 Comparison of regression results of formula parameters

Fig. 2 Test data and regression curves

Fig. 3 Memberships of test data

3 Rationality Verification of Fuzzy Nonlinear Regression Method

Eq.(2) can be expressed as a general nonlin?ear regression model

where y is an observable n?dimensional random vector; x is a known n?dimensional vector; θ is the unknown p?dimensional model parameter vec?tor; f(x,θ) is second order differentiable to θ; ε is a random error vector, and it is assumed that ε～N(0,σ2I).

Ref.[15] gives the deviation calculation of least square estimate of parameters in the nonlin?ear regression model

Fig. 4 Deviation change ofα

It can be seen from Fig.4 that the influence of the seventh data on the deviation of α is much greater than other data. According to the least square principle, it can be judged that this set of data is an outlier generated during the experi?ment, and its influence on the estimate of model parameters should be reduced during fitting to obtain accurate parameter values of the vibra?tion attenuation formula. That is consistent with the membership results in the last iteration of the fuzzy nonlinear regression method, which verifies the rationality of the fuzzy nonlinear re?gression method to adjust the weight of the resid?ual square according to the membership of each test data.

4 Conclusions

Based on the theory of fuzzy mathematics,this paper derived a fuzzy nonlinear regression equation for Sadov’s vibration formula, and the calculation results were compared with the non?linear least square method. Finally, the rational?ity of the method was verified and the following conclusions can be drawn:

① The difference between the fuzzy nonlin?ear regression method and the nonlinear least square method is that the weight of residual square in the objective function can be adjusted according to the membership of each data. When the fuzziness of blasting engineering and the in?fluence of outliers are not considered, the fuzzy nonlinear regression method will be transformed into the nonlinear least square method. So, the latter can be regarded as a particular case of the fuzzy nonlinear regression method.

② The test data and regression curves shows that, compared with the nonlinear least square method, the fuzzy nonlinear regression curve fit?ted most data points better. And it does not ad?just the majority of the data’s residual weight,which mainly reduces the residual weight of the seventh data and its impact on the fitting results.

③ The deviation calculation of least square estimate of parameters in the nonlinear regres?sion model showed that the influence of the sev?enth data on the deviation of α is much greater than that of the other data, and its influence on the estimate of model parameters should be re?duced during fitting. That is consistent with the membership results in the last iteration of the fuzzy nonlinear regression method, which verifies the rationality of the fuzzy nonlinear regression method to adjust the weight of the residual square according to the membership of each test data.