Zoneless approaches

In zoneless approaches, the seismic activity rate is calculated using the declustered earthquake catalogue without delineating any seismogenic zones over which the activity rate will be assumed to remain constant. The idea is to allow the expression of the seismic catalogue without biasing the shape of the activity rate function. The dependence on location has a continuous variation (as opposed to forcing it to be constant over certain zones), while the dependence on magnitude does not necessarily follow the GR relation.

These methodologies use non-parametric density estimation, in which the objective is to find the density function from which a given sample derives, without specifying a priori a specific shape for the density function, such as a normal distribution or a Gamma one; instead, the shape of the distribution is FIG. 7. Plots of the GR b-value for two zonations of the Iberian Peninsula, courtesy of M. Crespo.

The most basic form of non-parametric density estimation is the histogram.

The space over which the sample is distributed is divided into cells that are usually uniform in shape and size, although this is not a formal requirement. For each cell, the number of elements (seismic events in this case) is computed, with the final function acquiring a constant value over each bin that is proportional to its number of events. An example of histograms constructed with events of a magnitude between 4 and 5 around the Iberian Peninsula is presented in Fig. 8. As mentioned above, the concept of the histogram was introduced to the study of seismic activity by Frankel in 1995 [27] for computing the GR a-value.

An improvement by Frankel with respect to the traditional formulation of the histogram with step functions was to centre a smooth function that decays with distance from the centre of the cell.

Another improvement with respect to the histogram is the naive estimation.

The method relies on centering a density function on each element of the sample (instead of on each cell), adding up all such functions and then normalizing their sum. The smooth function can be any unit density function. It was initially proposed by Fix and Hodges in 1951 [28], but a more recent and very clear description was provided by Silverman in 1986 [29]. The shape of the kernel function and its spatial extent have to be decided by the user, as is the case for the width of the bins in the histogram.

The mathematical definition of the density estimated with kernel functions is as follows:

n is the number of elements in the sample

H is the bandwidth, a measure of the separation between sample elements K is the kernel function

And x_i is the position of event i.

An example of kernel density estimation, constructed with the same events as in Fig. 8 (magnitude between 4 and 5 around the Iberian Peninsula), is presented in Fig. 9.

FIG. 8. Histograms for events around the Iberian Peninsula with magnitude M between 4.0

FIG. 9. Kernel estimations for events around the Iberian Peninsula with magnitude M_W between 4.0 and 5.0. The kernel bandwidths considered, from top to bottom, are 0.25° 0.5° and 1° respectively. Image courtesy of M. Crespo.

For generating a density, λ_k, of the seismic activity rate, two changes are introduced in Eq. (8):

— The normalization with respect to the number of events, n, is omitted, thus the result is expressed in terms of number of events.

— Each kernel function is divided by an effective period, T, so the density of events is expressed per unit time.

With the above two changes, the expression becomes:

l_k M

The kernel function, K, the effective detection period, T, and the bandwidth, H, are the three main parameters that influence the activity rate density.

The effective period, T, ensures that the function has the desired units of events/year, as is necessary for a seismic activity rate. Its value is such that the total number of events of the same type, divided by the effective period, yields the actual seismic activity rate. As noted in Eq. (9), each event can be assigned a different effective period, T, which makes the methodology very versatile. Typical event characteristics on which the effective period usually depends are the event magnitude, the time of occurrence and the type of epicentral location (onshore or offshore, and whether the area was populated at the time of occurrence), but other factors that affect the probability of detection can also be incorporated in the effective period.

The resulting activity rate density, λ_k, depends on location as well as magnitude through the bandwidth, H. As can be seen, it is a summation of kernel functions, K, placed on each event of the catalogue with coordinates x_i. Each function is weighted with an effective detection period, T; the normalization is achieved through the bandwidth, H, which depends on the distance between events. The fact of dividing by H² ensures that the activity rate is in units of events·km⁻²·year⁻¹.

Several kernel functions have been proposed for use in seismicity modelling, specifically the Gaussian kernel, the inverse bi-quadratic kernel and a finite kernel that vanishes at distances beyond one bandwidth. These three types of kernels are presented in Fig. 10, although this list is not comprehensive and other types might be possible.