Background filters in the hazard app

The new background filters have been added to the Hazard Assessment application. The time of day filter can be used to see the effect of removing events during blasting/shift change on the hazard results. You can either view the results in raw or normalised form. The hazard calculations do normalisation for the event rate calcs anyway, to represent hazard in yearly terms. If your analysis period is 6 months, the number of events is doubled to represent a year’s worth of events. When applying the time-of-day filter though, the actual analysis period is less than 6 months, because several hours per day have been removed. Without normalisation, the hazard should always drop when applying the time-of-day filter, because you are removing events, and nothing else changes (i.e. still using 6 months). If normalisation is turned on, the time period that has been removed is accounted for in the hazard calculations. The results then represent accurately the state of the hazard during the relevant times of day.

Normalisation also applies to the short-term responses filter, where events can be removed based on a time and distance from a blast or significant event. In this case the normalisation is a bit more complicated. With the time-of-day filter, the effective analysis period is the same for the whole grid. In this case however, there will be an uneven distribution of space and time removed from the analysis. So, each individual cell has its own effective analysis period, based on how many triggers (and responses) are nearby. The idea is still the same though, without normalisation, the hazard will drop due to the removal of events without adjusting the analysis period. With normalisation turned on, the results will represent the hazard state outside of short-term response regions.

A new chart has been added to the Hazard Assessment app that shows the effect of different short-term response filtering on hazard. The chart works in a similar way as the Track Volumes over Time chart, by computing the hazard over and over again, automatically changing variables with each run. The chart and associated control panel can be found in the Hazard Assessment / Hazard ISO’s window, under the Response Analysis menu. To generate the chart, you need to specify a maximum response time, a time delta, and response distances (up to 6). The hazard will be calculated for each response distance and for each response time from zero to the maximum (at delta intervals). The hazard recorded is the probability of exceeding the design magnitude within the chosen grid, which is the value displayed in the footer of the 3D ISO view. It can take some time to calculate, depending on how many iterations you specify. The video below shows the chart being generated for response times up to 72 hours and response distances of 50, 100 and 150 m.

 

Stochastic Declustering Explained

As mentioned in the last blog post, a stochastic declustering algorithm has been implemented in mXrap to separate events into ‘clustered’ and ‘background’ components. It can be useful when designing seismic exclusions and re-entry procedures to separate seismicity that occurs in short bursts from seismicity that has low variability in space and time. Short-term exclusions cannot be used to manage the risk associated with background seismicity, since the hazard inside a potential exclusion would be the same as outside the exclusion. Efficient exclusion and re-entry procedures target areas where seismicity is most clustered and where the seismic hazard to which people are exposed can be reduced with a short disruption to production.

The filter controls for stochastic declustering in General Analysis are in ‘Event Filters / Background Activity’ and a new chart has been added to show the cumulative events of the two components in ‘Charts / Time Series / Declustered CNE’. An example of the cumulative declustered events chart is shown below for a week’s worth of events at the Tasmania mine. In this case approximately 32 % of events have been classified as ‘background’.

 

 

The declustering is based on the distribution of inter-event times (time between successive events). The distribution (PDF) of inter-event times has been shown to follow the gamma distribution (Corral 2004). The chart below shows how the events in the example above (black crosses) closely follow the gamma distribution (red line).  Hainzl et al. (2006) showed how to estimate the rate of background events from the gamma distribution, based on the mean (µ) and standard deviation (σ).

Background Proportion = µ2 / σ2

Background seismicity is generally assumed to be stationary and Poissonian. In other words, the average time between events is constant and known, but the exact timing between events is random. Each event is assumed to be independent and not affect the occurrence of other events. The inter-event time of a Poisson process follows the exponential distribution (green line).

The event distribution clearly deviates from the background distribution for small inter-event times. This deviation is caused by the clustered component of seismicity. The distribution of small inter-event times corresponds to the inverse distribution (yellow line), which is explained by sequences that follow the Modified Omori Law (MOL). In this case the slope of the distribution corresponds to the MOL with decay parameter, p ≈ 0.8.

The declustering method was described by van Stiphout et al. (2012). The probability that an event is part of the background (purple line) is calculated based on the inter-event time and the ratio between the background and gamma PDF’s. Events with small inter-event times are more likely to be clustered events. Events with large inter-event times are more likely to be background events.

 

 

It is important to note the random component in the declustering process. Each specific event may be classed as either ‘clustered’ or ‘background’ each time you run the declustering, although the overall proportions will remain the same (hence the ‘stochastic’ in stochastic declustering). There is also no consideration given to the spatial clustering of events, all events are assessed together in the time domain. There is also no consideration given to the magnitude of events.

The rate of background events is assumed to be constant although in reality the background rate will slowly vary over time, related to changes in system sensitivity, general rates of extraction and different mining locations. To account for long-term fluctuations in background rate, events are broken down into groups, and the background proportion is computed separately for each group. Groups of events are kept as small as possible, with a minimum number of events and minimum time period (user defined). The background rate is constant within each group.

Aside from General Analysis, the stochastic declustering process has been added to the Hazard Assessment, Short-term Response Analysis, and Seismic Monitoring apps. The background filters in the hazard app can be used to compare the seismic hazard of clustered and background seismicity (as per below). Background rates are also calculated for triggers in the short-term responses app and for the reference rate in the activity rate monitoring window.

 

 

For those wishing to read more about the declustering process, the CORSSA article by van Stiphout et al. (2012) is a good summary of many different approaches used in earthquake seismology, including the method described here.

New b-value isosurfaces

The isosurfaces for b-value have been upgraded in the latest root. There is now much more control over the isosurface levels. Up to 5 iso’s can be plotted for user defined ranges. A linguistic name can be assigned to each level and displayed in the legend. A new video has been uploaded to the Hazard Assessment page that explains the new b-value isosurfaces.

 

Hazard Assessment App – Excavation View

The grid-based hazard calculations in the Hazard Assessment app were discussed in a previous post. The Iso View describes the hazard at all locations within the mine but when you are considering the seismic risk for a particular work area, large events and strong ground motions may come from multiple sources. The Excavation View estimates the seismic hazard associated with working areas (minode locations) in a few different ways as described below.

P [ ML within R ]

The P [ML within R] marker style is the probability of exceeding the design magnitude within the design distance (R) of the minode location (per year). This is the simplest minode hazard estimate and you will notice the other marker styles take a bit longer to calculate because they include more complex ground motion (PPV’) calculations.

As discussed in a previous post, the grid-based hazard calculations result in a probability of exceedance assigned to each grid volume. So, for a design magnitude of ML2, the annualised probability of exceeding ML2 within each grid cell volume is computed. To compute the probability of exceeding ML2 within R of a minode, the exceedance probabilities for all the grid cells within R are integrated together.

In the 2D example below, there are seven grid cells within R of the minode. Let’s say each of these grid cells has a probability of exceeding ML2 of 1 %. Then, the probability of exceeding ML2 within R of the minode would be:

P [ML2 within R] = 1 – (1 – 0.01)7 = 6.8 %

The hazard for a single minode doesn’t help you much. You really want to know the hazard for the whole mine or for any possible work area. To compute the probability of exceeding ML2 within R of multiple minodes, you just integrate all the grid cell probabilities within R of any minode. This is illustrated below for a tunnel length with three minodes. Each grid cell is only counted once. In the hazard app, the footer of the Excavation View shows the P[ML within R] for all minodes. You can also use the selections tool to select any combination of minodes and the integrated hazard will be shown in the footer. The Volume Hazard tool calculates the P[ML within R] for any minode within each of your filter volumes.

As you can see from the illustrations above, the grid cells that are included “within R” are based on the distance to the grid centre point. This is why we recommend using a value for R that is larger than the grid spacing you are using. If you specify R much less than the grid spacing, you may see some odd artefacts where some minodes have much fewer grid points associated than others.

Ground Motion Hazard

The other minode marker styles in the Excavation View express the hazard based on the probabilistic strong ground motion (PPV’). As discussed in the last blog post, a Strong Ground Motion relationship is required to calculate the probability distribution of PPV’ based on the event source details and R. For each minode, the ground motion hazard, P[PPV’], is the probability of exceeding the design PPV’, anywhere within the volume associated with the minode (the tunnel section, per year).

For each minode, there is a probability of exceeding the design PPV’ due to a large event occurring within any particular grid cell. To compute P[PPV’] for a minode, each grid point must be considered and the probabilistic contribution of each integrated together. The grid cells closest to the minode (smallest R) are the most likely to contribute the most to the minode P[PPV’] but high hazard grid cells will have a larger zone of influence because of the potential for very large events.

The grid-minode combinations with a very small P[PPV’] are ignored in the analysis to speed up the calculation. The minode P[PPV’] is calculated from the remaining grid point combinations. For each minode, the threshold probability to ignore a grid point can be specified from the control panel. Increasing the threshold is slightly less accurate but will increase calculation speed.

The P[PPV’] for each grid-minode combination is computed by discretising the magnitude, distance and PPV’ probability distributions. From the grid-based seismic hazard calculations, the magnitude distribution is known and the probability of each magnitude bin of large events can be calculated. Then, the distance from the grid cell to the minode volume is represented by the R distribution in a similar way to the figure below.

From every combination of ML and R bin, the strong ground motion relationship is used to calculate the full PPV’ probability curve for the minode. Only the magnitude bins that can possibly result in a PPV of interest are included in the calculations. This is controlled by the PPVmin control parameter. The PPV’ probability curve is plotted above PPVmin. The P[PPV’] can be quickly calculated for any design PPV’ values above PPVmin from the probability curve.

Note the example PPV’ probability curves in the figure below. The RED line is the probability curve for a single minode but as with the P[ML within R] calcs discussed above, you will often want to evaluate larger tunnel areas, or even the whole mine (BLUE). The PURPLE line is the probability curve for a small section of tunnel with multiple minodes and indicates the yearly probability of exceedance for PPV’ anywhere along that tunnel section. The GREEN line is the probability curve for the whole mining level.

Clearly the probability of exceedance will tend to increase as longer sections of tunnel are considered. So, even though the P[PPV’] for a single minode is small, the accumulated hazard when considering the whole mine can be much higher. This is why the hazard app displays the ground motion hazard in a couple of different ways other than the individual minode P[PPV’] to make the hazard ratings more intuitive.

Equi-probability zones

The equi-probability zones marker style is a simple ranking of minodes from lowest to highest P[PPV’]. The marker style value is a percentile rather than a probability. For example, let’s say you have 100 minodes, the top 5 minodes with the highest hazard will be red (0.95-1.00). The next top 5 (6-10) highest hazard minodes would be orange (0.9-0.95) etc.

Note that the equi-probability zones do not illustrate an absolute hazard rating, rather illustrates relatively low hazard areas and relatively high hazard areas. There may be cases where the top 5% of minodes are actually low hazard. Conversely, the bottom 5% minodes may still be quite a high absolute hazard. The PPV probability chart plots the curves for each equi-probability zone.

Cum P[PPV]

The Cumulative P[PPV] marker style also ranks minodes from lowest to highest P[PPV’] but the marker values are probabilities accumulated from lowest to highest. As described previously, to accumulate the hazard for multiple minodes, the probabilities are integrated together. For example, if your have four minodes with individual P[PPV’] of 1%, 2%, 5% and 7%, then the probability of exceeding the design PPV’ AT ANY of the four minodes is:

P[PPV’] = 1 – (1 – 0.01)(1 – 0.02)(1 – 0.05)(1 – 0.07) = 14.3 %

The Cumulative P[PPV] marker accumulates the probabilities in the same way, one at a time, from lowest to highest hazard. So for example:

  • The lowest hazard minode has the same individual P[PPV] and Cum P[PPV].
  • The second lowest hazard minode has a Cum P[PPV] equal to the combined hazard for the two lowest minodes.
  • The minode with the median individual hazard has a Cum P[PPV] equal to the combined hazard integrating all lower 50% of minodes.
  • The highest hazard minode has a Cum P[PPV] equal to the accumulated hazard for all minodes.

The Cum P[PPV] marker has the benefit of showing both the cold-to-hot scale of all minodes as well as an indication of the absolute hazard.

Prob Class

The Probability Class marker style is similar to the Cumulative P[PPV] marker style, except you can specify your own colour scale and classes and give them a name. This can help to communicate changing hazard areas to operational personnel.

You can see the classes in the risk matrix. You should notice that the Cum P[PPV] corresponds to Prob Classes according to the numbers in that matrix.

For more details on the ground motion hazard calculations see Wesseloo (2018).

Strong Ground Motion Relationship

The Strong Ground Motion (SGM) relationship is used to calculate the Peak Particle Velocity (PPV) generated by a seismic event. You may also hear this referred to as a Ground Motion Prediction Equation (GMPE), but only the maximum velocity is estimated, i.e. the strong ground motion, rather than the full, complex wave motion.

The PPV is generally calculated for a specific location based on:

•  distance to the seismic event (R)

•  source magnitude (ML)

•  source radius (Rs)

•  static stress drop (SSD)

The source radius can be computed as a function of magnitude and the adjustment due to SSD is sometimes excluded. So PPV is often simply a function of ML and R.

You have probably seen the SGM relationship illustrated in a similar way to the figure below. What is sometimes not recognised is there is an associated uncertainty in these relationships. In the case below, the PPV values are based on a 10% chance of exceedance for a given ML and R.

The wave motion from a seismic event through the rock mass is highly complex and uncertain. So, for a given ML and R, the PPV is not a single value but a probability distribution. This is illustrated below, for a ML2 at a distance of 100 m, the PPV distribution is plotted along with the 10th, 50th and 90th percentile values for probability of exceedance.

 

 

There are many factors that contribute to the uncertainty in ground motion that results from a seismic event. The ground motion does not radiate from a seismic event uniformly. Each source mechanism has its own radiation pattern where the magnitude of the ground motion varies depending on the direction. The radiation pattern also differs for the P-wave and S-wave.

As the body waves radiate outwards from the source they attenuate but rock mass anisotropy affects the rate of attenuation. The wave attenuates faster when cutting through lamination than it does when travelling along (parallel) to the bedding or foliation plane.

Excavations, different lithologies and major contacts and discontinuities will create reflections and refractions. The wave will split into a separate P-wave and S-wave when it reflects off a boundary or refracts into a new medium. Multiple waves can superimpose to create stronger ground motions that the individual waves.

 

Surface effects

The SGM relationship does not give the PPV expected on the excavation surface. The SGM equation is based on the recorded PPV at sensors which are normally installed well into the rock mass, away from the excavation surface. So, the calculated PPV includes the uncertainty from different radiation patterns, natural variability, reflections and refractions but does not include surface effects. The strong ground motion of the body wave, without including the effect of the excavation surface, is sometimes referred to as PPV’. The true surface PPV is the PPV’ with an additional amplification factor applied.

The amplification is due to a couple of different effects of the surface. The amplitude is expected to double at the free surface due to the superimposition of the incoming wave and the reflected wave. The amplification is more than double for a corner pillar due to the closed geometry. The body waves can also interact with the free surface to form Rayleigh and Love surface waves. These waves propagate along the surface rather than through the rock mass. The low velocity fractured zone around the excavation can enhance the formation of surface waves as the seismic energy is trapped between the free surface and the fracturing boundary. Waves also increase in amplitude as they move into a lower velocity medium such as the fractured zone.

In earthquakes, surface waves cause the most damage to infrastructure. The amplitude of Rayleigh and Love waves tend to be higher than body waves. Surface waves also attenuate more slowly, i.e. travel further, since the geometric attenuation is only along the surface rather than all three dimensions. Another common observation with earthquake damage is that buildings on soft soils are more heavily damaged than buildings on solid rock. This is a similar case as the fractured zone around excavations although there is less experimental evidence of this phenomenon in underground tunnels.

If you are interested in reading more, these papers are a good place to start:

•  Potvin and Wesseloo (2013) – Includes a discussion of factors influencing dynamic demand on ground support.

•  Wesseloo (2018) – Description of the SGM relationship and hazard calculations done in mXrap.

 

Procedure for site specific SGM calibration

The SGM relationship is used to calculate PPV in the Hazard Assessment and Large Event Analysis apps. The default relationship is from the Canadian Rockburst Support Handbook. This relationship is mostly based on recorded ground motions at Brunswick, El Teniente and Creighton mines and may not be applicable to your site. It is fairly simple to calibrate your own site specific SGM relationship using the data recorded by your seismic system. The PPV is recorded by each sensor for each event. With this data, we have a tool to calibrate your site specific SGM relationship. If you would like to do this for your site, the procedure is as follows:

1.  Export your PPV data from your seismic database. We have done this a few times for IMS data. It is called an Event-Trigger query that is a table with a row for each sensor hit per event. Contact support@mxrap.com if you need assistance. At minimum we need the export to include the sensor location, sensor PPV, event location, event magnitude and event static stress drop.

2.  Save the PPV data into the #Data folder in your root and run a default backup in mXsync. Contact support@mxrap.com to let us know you would like us to calibrate your SGM relationship.

3.  We will generate your site specific SGM equation and send the info back as a patch in mXsync.

4.  Apply the patch in mXsync and run another default backup.

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