The a/b value is sometimes used as a measure of seismic hazard but there are some common mistakes made with this analysis and interpretation.
What is a/b?
The Gutenberg-Richter distribution is a statistical model that describes a log-linear relationship between the number of events, N, exceeding magnitude, M.
log10 N = a – bM
At N = 1, M = a/b. The figure below shows an example of a frequency-magnitude chart with the a/b value highlighted.
Does a/b mean anything?
It is important to distinguish between properties of the dataset and properties of the statistical model. The a/b value is a property of the Gutenberg-Richter statistical model but it is defined at a particular data point (N = 1). The a/b value does have some meaning, but that’s really only because the a and b value both mean something (although I’ll come back to the a-value later). In terms of seismic hazard, the activity rate and b-value are the two primary inputs required.
The focus on the magnitude where N = 1 is somewhat arbitrary. The statistical model describes the relative frequency for all magnitudes. It is just as valid to normalise the frequency axis to a percentage i.e. express N as a percentage of the number of events at M = Mmin. So in the figure below, at Mmin, the frequency is 100% and events over M = 1 represent 0.1% of all events over Mmin. Note the a/b magnitude represents approx 0.006% of events. So the magnitude at N = 1 loses its significance. Asking what is the significance of a/b is like asking the significance of the magnitude of the top 0.1% of events? Why not the top 0.01% or 0.001%?
The normalisation trap (or the non-normalisation trap)
The reason the a/b value doesn’t mean much for seismic hazard is because the a-value by itself is meaningless. The number of events, by itself, doesn’t tell you anything about hazard because it has no associated time and space units. It should be pretty easy to understand the importance of normalisation to regular time and space units. If I tell you there has been 100 events, you don’t know anything about what seismic hazard that represents. It could be 100 events in a very small volume, in a very small time period; this would be a high hazard. It could be 100 events in a very large volume over a very long time period; this would be a low hazard. So the important thing for seismic hazard estimates is the event rate density, i.e. the number of events, per unit time, per unit volume. Only then can you compare apples with apples.
One final point. A constant event rate density, and a constant b-value over time represents a constant hazard state. The problem is that the a/b value without normalisation is entirely dependent on how long you have recorded this constant hazard state. The total number of events (i.e. the a-value) continuously grows and so does the a/b value, even though the hazard state is not changing. This is why without normalisation, the a/b is not a measure of hazard.
If you normalise the event count based on the event rate density and a standard time and volume, the a/b value can be a measure of hazard. However, in terms of probabilistic seismic hazard, the probability that the largest event in the database will exceed the a/b value is ≈ 63%, assuming an open-ended Gutenberg-Richter distribution or a very high MUL (MUL >> a/b).
- The a/b value is a property of the Gutenberg-Richter model, not of the dataset
- There is no special significance to the magnitude where the Gutenberg-Richter model crosses N = 1
- The a/b value is a function of the number of events
- Without space and time information, the a/b value (and the a-value) are not indicative of hazard
- When comparing different times and zones using a/b, you must normalise using the event rate density and a standard time and volume
- The probability of the largest event exceeding a/b is ≈ 63%
When you are using the Frequency-Magnitude chart, it can be easy to forget it is log scale and this can distort a few things. Consider the chart below, have you ever thought the Gutenberg-Richter distribution doesn’t look right? Think it isn’t matching the large events very well?
The Gutenberg-Richter distribution is a statistical model of the data. Consider what the chart looks like in linear scale rather than log scale. The difference at the tail of the distribution (largest events) seems much less significant right? The other interesting point is the relative proportion of events above and below the Mmin. There is roughly only 20% of events in you database that are above the magnitude of completeness.
Obviously in linear scale, you can’t see what’s happening at the tail very well, that’s why we use the log scale in the first place :)
There are many reasons you might want to store a short snippet of text associated with an event. There are two ways to do this in mXrap; event tags and event comments.
Event tags can be used to group events into categories. Example tags might be “suspected blast”, “damage occurred”, “suspect location”, “outlier” or “likely crusher noise”. These tags can be used in event filters to quickly show or hide particular categories.
Event comments are a second option to assign user text to events. Each event comment can be unique and about anything. They have no effect on event filters.
You can find videos on “Event tags” and “Event comments” at the training video page below. Both event tags and comments are shown in the main events table in General Analysis.
The event tags system has been modified recently. If your mXrap looks different to the video, you might need a root update. This process is now quick and easy with mXsync. We just need 5-10 minutes to connect via teamviewer / webex / gotomeeting.
Contact firstname.lastname@example.org for assistance.
You can use selections to filter events in General Analysis. This gives you a lot more freedom than being restricted to the traditional min/max range filters. Follow the steps below to see how you can use this feature to plot the Frequency-Magnitude chart for events occurring during periods of high apparent stress.
You can also check out this page to watch the “Selection boxes” and “How to use selections in the base filter” videos.
Step 1 – Create a new selection on the Apparent Stress Time History chart. Note that selections can be made in any 3D view, chart or table in a similar way.
Step 2 – To apply the selected events to the filter, go to the Events Ranges panel, hit “Copy selections to Base filter” and switch on the selection filter below. Now only the selected events will be used in the Frequency-Magnitude chart.
Step 3 – To turn off the blue selection icons, go to the Events series and turn off the “Highlight selected” option.
Note that if you adjust the selection box or make another selection, you need to hit the “Copy selections” button again to apply the changes. Use the switch to turn off the selection filter and return to original filter.
In the case of the Apparent Stress Time History chart, selection boxes are applied to the frequency line rather than the events in the background. The series that is active for selection can be modified. Look for the “Select” option in the series controls on the right-hand panel.
The distance to survey filter has been around for a while but now we have added a couple of new charts to further investigate the relationship between seismicity and your input surveys. The charts have been added to the General Analysis application, under the Charts menu, look for Distance to surveys.
There is a training video to explain a few things about the charts, find it here: https://mxrap.com/training-videos/dist-surveys/
The example below shows how you can compare your seismicity around your geological structures. Of course, sometimes a structure will have a lot of seismicity nearby because you happen to do a lot of blasting nearby. You can also plot blasts as a function of distance to survey!
If you can’t see the charts in your mXrap, you might need a quick root update, ping us an email at email@example.com and we’ll sort it out. Now we’ve setup everyone with mXsync, updates are very easy, will probably only take 10 minutes on teamviewer/webex :)
What do you think of the new charts? Any ideas for additions or improvements? Contact support and let us know!