The Grid Based Analysis application can be used to evaluate the spatial distribution of various seismic parameters. There are a range of source parameter options available, and they can give indications to the rock mass behaviour. Some parameters can be considered as a proxy (stand-in) for rock mass stress, while other parameters can be a proxy for the amount of deformation. There are also parameters available that are associated with the rock mass mechanism or event type.
In grid-based analysis, a representative value of each seismic parameter is assigned to each grid point based on nearby events. This post summarises the calculation methods and control parameters used to assign each seismic parameter to the grid.
Average vs. Cumulative Parameters
Most of the source parameters in the grid-based analysis app are what we call average or cumulative parameters. This is a distinction both in the nature of the parameter and the underlying calculation method. For some parameters, particularly those associated with rock mass deformation, it makes sense to find the cumulative effect of each event. Like Moment for example, the salient information is how much deformation there has been in a certain area. For other parameters, generally related to stress (like Energy Index), the cumulative value doesn’t really have much meaning. This is where we are interested in the average value instead. For parameters that scale exponentially, the average is calculated for log10(parameter).
The search radius
Grid-based analysis is fundamentally about associating nearby events with grid points. The event‑grid association is made within a certain search radius, which is defined differently for average parameters and cumulative parameters.
For average parameters, a different search radius is used for each grid point, based on the event density nearby. There are three parameters to control the search radius around each grid point:
- Rmin. Minimum search radius. All events within this distance from the grid point are assigned, no matter how many there are. Default: 2 x Grid Spacing.
- Search N. The search radius is expanded until at least these many events are within range. Default: 50 events.
- Rmax. Maximum search radius. The search radius does not expand beyond this range, even if the Search N has not been reached. Default: 8 x Grid Spacing.
A density-based quality check is applied to all grid cells. At least 10 events must be found within a certain threshold distance of the grid point. The default distance is 90 (in native units). If there are less than 10 events within this range, no value is given to that grid point.
For cumulative parameters, rather than defining a search radius for each grid point, the search radius is different for each event, depending on the source size (source radius). No matter what the source radius is for an event, the search radius will not be set below 1.5 times the grid spacing, or above 200 m. The radius is converted to feet if that is the native spatial unit.
The kernel function
The search radius defines which events are associated with the grid but parameters are weighted depending on how close the event is to the grid point. The kernel function is what defines this distance weighting.
The kernel function order defines the shape of the kernel function. Various kernel functions are plotted on the right for kernel orders between 0.3 and 50. The default kernel function order is 3.
For average parameters, the inverse-distance weighted mean is calculated using the specified kernel function. For cumulative parameters, the kernel function controls how each event is distributed onto the grid cells. The contribution to each grid point is adjusted to ensure the cumulative parameters are conserved. In other words, if you compute the grid using 100 events or 1kJ of energy, the final result on the grid should also add up to 100 events or 1kJ.
Shift Change / Blast response ratio
The response ratio for blasts and shift change is based on the event time of day. Blast and shift change periods are defined in General Setup Windows/Grid-based Analysis Settings. The response ratio is the activity rate inside the specified time-of-day periods divided by the activity rate outside those periods.
A high blast response ratio, for example, means there are a lot of events associated with blasting, and much less activity throughout the rest of the day. A blast response ratio of below one indicates the activity rate is higher during production periods, which is a sign of ore pass or other production noise dominating the dataset.
The search radius for calculating the response ratios are the same as the average parameters; a minimum number of events within a minimum and maximum radius. The difference is that there is no distance weighting applied, all events inside the search distance are treated the same.
The event-grid association for the b-value calculations is the same as for the blast and shift change response ratios, i.e. not weighted by distance. A decision metric is used to find the Mmin (magnitude of completeness) for each grid point. Once the Mmin is known, the b-value can be calculated based on the average magnitude above Mmin.
The decision metric to find Mmin has three components:
- Log10(k). k is the number of events above Mmin. This is part of the decision metric so that as many events are found above Mmin as possible, and to push the solution away from the distribution tail, where there are few events and the other parameters are erratic.
- b. The b-value is part of the decision metric to avoid over-shooting the Mmin. Higher b-values are given more weight since underestimating Mmin would result in a lower b-value.
- 1 – KS. The KS value is a measure of the goodness of fit between the data and the Gutenberg-Richter model.
An example of the decision metric calculation is illustrated in the figure below, along with each of the three components. The metric is calculated for each event, as if it were the Mmin. To speed things up in the grid calculations, the metric is evaluated in bins, usually 0.1 magnitude units so you will generally only get Mmin to one decimal place in the results. The final Mmin is when the decision metric is at its maximum value.