Sand Box: Construct and evaluate magnitude-rate distribution scenarios
Active seismic regions release their tectonic strain mainly through earthquakes of different magnitudes on faults and subduction zones. For hazard analysis, it is necessary to quantify the rates of occurrences of these different magnitude earthquakes on different regional seismic sources. Seismologists call this process building magnitude-rate distributions (MRDs) for seismic sources. The most common MRD used in practice is the Gutenberg-Richter (GR) distribution developed by Charles Francis Richter and Beno Gutenberg in 1944 based on California earthquake data.
It is a fact of nature that large-magnitude earthquakes occur less frequently than small ones. In other words, the rates of occurrences of earthquakes decrease with increasing magnitude.
$$n=-\frac{dN}{dM}… (1)$$
Where \( \frac{dN}{dM} \) is the change in the rate of occurrence per unit magnitude identified by \(n\), known as the density function. Note that \(N\) for any given magnitude \(M_{ref}\) defines the occurrence rate of earthquakes of magnitudes \(M \ge M_{ref}\). The negative sign signifies that the rates of earthquakes decrease with the increasing magnitude.
The GR distribution defines the density function as
$$n=10^{a-b * M}…(2)$$
Where \(a\) and \(b\) are region-dependent constants that control the level of seismicity and the change in the rate of occurrence with magnitude, respectively. Using equations 1 and 2, the total number of earthquakes within the magnitude range of \(dM\) can be estimated as
$$dN= 10^{a-b * M}*dM…(3)$$
Figure 1 shows a plot of \(log(n)=a-b*M\) as a function of magnitude. The \(b-value\) defines the slope of the curve, which characterizes the relative rates of occurrences of large and small-magnitude earthquakes. Sources with low \(b-value\) release more strain energy by large magnitude earthquakes than those with high \(b-value\). The typical \(b-value\) for most regions is within \(0.8 – 1.2\).
Figure 1.
The \(a-value\) defines the scale of seismicity and can be estimated by the value of \(n\) at \(M = 0\) in equation 3. For example, in the last two US national seismic hazard models, USGS reported background seismicity at grids as the rate of earthquakes of magnitude \(-0.05 \le M \le 0.05\) (\(M = 0 \pm 0.05\)). Accordingly, using equation 3, the \(a-value\) can be estimated as \(a=log(N_{0 \pm 0.05} /0.1)\). One can fully construct the GR magnitude-rate density distributions for all grids using the estimated (a-value) in conjunction with the (b-value) and the upper bound magnitude provided by the USGS.
The occurrence rate of earthquakes of magnitudes \(\ge M\) can be estimated as
$$N=-\int_{M}^{M_{max}} {n*dM}…(4)$$
Using the formulation of the density function from Equation 2, the cumulative rate can be formulated as
$$N=N_0*\frac{{10^{-b*(M-M_0)}-10^{-b*(M_{max}-M_0)}}}{{1-10^{-b*(M_{max}-M_0} )}}…(5)$$
where \(N_0\) is the rate of earthquakes of magnitude \(M \ge M_0\) and \(M_{max}\) is the maximum magnitude considered for the GR distribution.
Figure 2. shows the plot of the cumulative magnitude-rate distribution corresponding to the density function in Figure 1. Also shown in these figures is the visual correspondence between the occurrence rate for \(M=6.5 \pm dM/2\) earthquakes using density and cumulative distributions.
Figure 2.
Equation 5 and the plot in Figure 2 are known as the truncated MRD. It is called truncated because the MRD is normalized so that as the magnitudes of earthquakes increase, the rates of occurrence decrease and go to zero at the maximum magnitude.