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.

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 M_{L}2, the annualised probability of exceeding M_{L}2 within each grid cell volume is computed. To compute the probability of exceeding M_{L}2 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 M_{L}2 of 1 %. Then, the probability of exceeding M_{L}2
within R of the minode would be:

P [M_{L}2 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 M_{L}2 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 PPV_{min} control parameter. The PPV’ probability curve is plotted above PPV_{min}. The P[PPV’] can be quickly calculated for any design PPV’ values above PPV_{min }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:

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).

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 (M_{L})

• source radius (R_{s})

• 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 M_{L} 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 M_{L} and R.

The wave motion from a seismic event through the rock mass is highly complex and uncertain. So, for a given M_{L} and R, the PPV is not a single value but a probability distribution. This is illustrated below, for a M_{L}2 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.

The Iso View in the Hazard Assessment application expresses the seismic hazard in two ways.

The current yearly hazard within the chosen grid volume. This is shown in the footer of the 3D view, as the probability of an event exceeding the design magnitude.

The spatial distribution of the hazard. This is highlighted by the hazard isosurfaces.

In the case below, the design magnitude is set as M_{L}2. The corresponding hazard isosurfaces for M_{L}2 can be interpreted as the most likely location for that event to occur.

The M_{L} rating essentially delineates the areas of the mine from lowest to highest hazard. The volume bounded by the M_{L}2 isosurface indicates the M_{L} rating is above M_{L}2. Note that the colours in the legend are slightly different than the isosurfaces’ apparent colour in the 3D view. This is due to transparency effects and viewing multiple transparent surfaces on top of one another.

It is important to note that while the data period can change (6 months in the example above), the hazard calculations are all referring to the yearly hazard. This is a simple matter of normalisation. E.g. if you record 100 events in an area in six months, this area is assigned an activity rate of 200 events per year.

The use of yearly hazard is to help interpretation. Reducing the time period used in the definition reduces the probabilistic hazard and this can be misleading. For example, let’s say you give your mine manager a report every day and it says that based on recent data, the probability that we will experience an event in the next 24 hours over M_{L}2 is 0.77%. You do this every day for a year and each day, the mine manager looks at the number and thinks, “Hmm, 0.77%, that’s pretty small, risk is pretty low”. A daily hazard of 0.77% is the same as the yearly hazard in the example above.

1 – (1 – 0.0077)^{365} = 94%

The mine manager may interpret the risk more accurately when presented with the same hazard but expressed for a hazard period that is more intuitive.

The current yearly hazard displayed in the footer of the 3D view applies to the entire volume of the chosen grid. We also compute the yearly hazard in the VTM table in General Analysis. So, you might reasonably assume that if you specify a volume in General Analysis the same as the grid volume in the Hazard app, the two numbers should match. In fact, while the probability of exceeding M_{L}2 is 94% in the example above, the same volume and time period in the VTM table gives 86%.

This is because the two calculation methods are quite different. To compute hazard, the main inputs are the seismic activity rate, and the b-value (M_{min} and M_{UL} are also required). In the VTM table, a single b-value and activity rate is computed for events within the volume, and the seismic hazard is computed directly. In cases where the b-value does not vary significantly within the volume, this is a reasonable approach. However, in most cases, the b-value varies in space, and this approach tends to underestimate the seismic hazard.

This is illustrated in the figure below. You can represent the full volume with its activity rate and b-value to compute the probabilistic hazard, like in the VTM table. In the Hazard app, the variations in activity rate and b-values are calculated on a regular grid through space (in sub-volumes). While the event search radius for each grid point may exceed the grid cell spacing, the activity rate is normalised and the b-value is assigned to represent the seismicity for the specific grid cell volume. The probability of exceeding the design magnitude within each sub-volume can then be calculated. Then the probabilistic hazard for the full volume can be calculated by integrating together all of the sub-probabilities.

M_{L} Rating – Technical Meaning

As mentioned already, the yearly seismic hazard is expressed as the probability of exceeding the design magnitude. An alternate definition of hazard, is to use a design reliability rather than a design magnitude. I.e. the hazard can be expressed as the magnitude that, to the design reliability, will not be exceeded. We use a reliability of 85%. The M_{L} rating is the design magnitude that would have a probability of exceedance of 15%.

An M_{L} rating is assigned to each grid point to compute the isosurfaces. On the surface of the M_{L}2 iso for example, the M_{L} rating refers to the magnitude that, to a reliability of 85%, would not be exceeded within the standard volume given one year’s seismicity. The standard volume we use is that of a sphere of 50m radius.

We have added some new features to the Hazard Assessment app to calculate the minode hazard for filter volumes. This works just like the current minode calculations, where you can select minodes and compute the probability, P of exceeding your design magnitude, within R of any selected minodes. The volume hazard refers to the seismic hazard for minodes within the filter volume. The same backdate, backrange, Mdesign and R parameters apply as the existing tools.

Another tool has been added to track the volume hazard over time. Essentially this repeats the volume hazard calculations, stepping the backdate through time and plotting the hazard per volume. Refer to the “Track Volume Hazard” training video for a walkthrough of the new tools in the hazard app.

We will need to upgrade your root before you can use the new tools. If you would like us to upgrade, drop an email to support@mxrap.com. Root upgrades are fairly quick but you will need to give us access via Teamviewer, Webex or similar.