Blasting

Monitoring Page 3.

Confidence Limits

The explanation so far has focussed on obtaining the best fit line, which will enable predictions of an average PPV level. This means 50% of blasts will be above that level and 50% will be below. What is usually required is a prediction of the PPV value based on a given Scaled Distance, which will only be exceeded on a certain number of occasions (e.g. 5%, giving 95% below that level). This can be determined using the Standard Error of the PPV data together with the appropriate multiplier (i.e. number of Standard Errors) which can be obtained from statistical tables.

In the past there has been some confusion as to which Standard Error and which multiplier should be used. The Standard Error is confusing because of the different terms which are often used for the same thing. These include the standard error of estimate, standard error of regression, estimated standard deviation of errors, etc. The Excel regression routine gives a table of results as shown in Table 3 (in Monitoring; Page 2), where there are a number of Standard Errors shown. The one that is required here is the upper one (0.9487), shown in the regression statistics, and not the two relating to the coefficients of the intercept and gradient.

The confusion over which multiplier to use arises because with a normal distribution, a confidence limit normally applies above and below the mean (twin-tailed). With blasting, however, a single-tailed test is used because the desire is to include everything below an upper limit; it doesn't matter how low the vibrations get.

Figures 11 & 12 show why a multiplier of 1.645 should be used to obtain the required 95% level. In programmes such as Microsoft Excel, the confidence limits which can be calculated during the regression analysis are for a twin-tail and would therefore give the wrong results. However, figure 11 shows that the correct result for a 95% level would be obtained if a value for 90% was put into the programme (i.e. the number of Standard Errors for a single-tail 95% level is the same as a twin-tail 90% level).


Figure 11. Single tail multiplier (1.645) for 95% level.	Figure 12. Incorrect use of 1.96 multiplier gives 95% for twin tail, but 97.5% for single tail.

The equation for the 95% PPV level (i.e. that level of vibration which for a given Scaled Distance will only be exceeded once in twenty blasts) is given as follows:

log(PPV) = b.log(d/√MIC) + log(a) + (1.645 x Standard Error),
where (d/√MIC) = Scaled Distance.

This equation shows that a value is being added to the intercept which means the 95% will always be higher than the best fit line.

Very often a 95% PPV level is set as a limit, in which case, the above equation can be rearranged to make the MIC the subject which will give the blasting engineer the maximum amount of explosive which should be detonated at any one time.

log(MIC) = 2.(log(d) - ((log(PPV) - log(a) - (1.645 x S.E.)) / b))

When the site factors, distance and 95% PPV limit are substituted into the equation, it will give the MAXIMUM instantaneous charge weight which can be used to ensure a 95% probability of being within the limit (i.e. one blast in twenty is predicted to be over the limit). If a lower MIC is used then there will be a greater probability of being below the PPV limit.

Figure 13 shows the best fit and 95% lines for a good data set, together with the Scaled Distances which are related to a PPV of 10mm/s. If the blast engineer wants to design a blast which produces an average (50% limit) PPV of 10mm/s, then the Scaled Distance of 14.8 would be used. Table 4 shows how this Scaled Distance results in different MICs depending on the distance between blast and monitoring location (SD = distance/√MIC, or rather MIC = (distance/SD)2). More usually, the blast will be designed to a 95% limit, which for 10mm/s gives a Scaled Distance of 21.4 with its own set of MICs for particular distances. These charge weights will of course be lower than the 50% limit as more blasts must fall below the 10mm/s level.

Figure 13. PPV & Scaled distance plot with SDs for best-fit and 95% lines.

Table 4. Table of charge weights for different distances.

	Scaled Distance = 14.8	Scaled Distance = 21.4
100m	45kg	25kg
250m	285kg	161kg
500m	1141kg	644kg

Figure 14 is an interactive version of Figure 13, but this time the limiting PPV is 6mm/s rather than 10mm/s. This figure also shows the effect that "spurious" data points can have on the results.

Figure 14. Interactive diagram of the effect of spurious recordings in the blast design.

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Figure 15 shows the distance√MIC curve, from which the maximum charge weight to ensure compliance can be obtained for a range of distances. This curve is simply the graphical representation of the data in Table 4.

The fact that the 95% line is being used for the predictions means that the data scatter (which can be quantified by the correlation coefficient and Standard Error) is already being considered. This is because the greater the scatter, the higher the Standard Error and the further away the 95% line will be from the best fit line, thus resulting in higher Scaled Distance for a given PPV and therefore lower permitted MICs.

Figure 15. Maximum charge weights for a range of distances, for SD=21.4.

What is now clear is that IF it is possible to predict PPV's very accurately, then the permitted charge weights will be higher. It is true that this will result in higher average PPVs, but 95% of the blasts will still be below the limit.

The example given is an extremely good data set with excellent predictability. The question arises, if we start with a mass of data which does not show a good regression line, can we legitimately reduce the data with the intention of improving the correlation coefficient and Standard Error, resulting in higher permitted MICs? The answer is that this can be achieved in a number of ways and these will be explored under good practice.

Very often the site factors are determined by means of a test blast before production begins. This will involve monitoring a number of shots, often single hole, and recording at a number of different locations with a broad range of distances. There are severe limitations with this which will again be considered in the next section.