# Method

We developed the Probability-based Magnitude of Completeness (PMC) method to asses the completeness of detection of seismic networks and to deliver a full probabilistic description of the capabilities of each station and the entire network.

## Input Data

To apply the PMC method, one needs to first collect the necessary input data. The main data input is the earthquake catalog of the respective network. This catalog needs to contain detailed information about the picks made at the stations of the network. From these pick information, the method computes the detection capabilities of each station of the network. Further needed is the magnitude definition (or attenuation relation) used at the network for the dependence of distances to magnitudes given amplitudes at the seismometers. To compile the detection probabilities of the entire network, the triggering condition of the network needs to be known. This condition describes the algorithm through which the location procedure is triggered. Usually, if the network detects strong signals at 4 different stations within a certain time frame, the location procedure is called and attempts to locate the event.

## Preprocessing

Station detection capabilities are computed from the pick history of each single station. A station's pick history includes information which earthquake has been picked at this station but also which event was missed (not picked). To compile this information, one needs to know when a particular station was in operation and delivering data to the processing center. Unfortunately, standard station databases at networks store only the dates of installation of a station and its dismissal. However, our experience shows that many stations are interrupted in their operation due to damages, malfunctions, lost connections to the processing center, etc. Therefore, in a first step, we try to estimate the real on- and off-times by investigating the changes in picking frequency of each station. If the inter-pick times become much larger than the average inter-pick time, such an period is considered an off-time of the station. Figure 1 shows the pick activity of an example station of the GeoNet in New Zealand and the on-times derived from the inter-pick times. As can be seen, there is one multi-year period in the beginning in which the station was likely not in operation (interrupted black line). The station operation was further interrupted by several shorter periods of inactivity.

**Fig. 1**: *Pick activity of an example station of the GeoNet in New Zealand. The x-axis indicates time and spans a period of approximately 10 years. The reported on- and off-time of the station is indicated by the gray box. Green dots show at which times the station picked events and the black bars indicate our estimate of the on-times of this station.*

## Station Capabilities

After estimating the on-times of each station, we compute the station detection capabilities. For each station we collect the information about which event was picked at the station and which event was missed. For each event we know the magnitude and the distance of its hypocenter to the station (Fig. 2 left). The set of this kind of magnitude/distance/pick information constitutes the so-called “raw” distribution (Fig. 2 bottom). Each green dot represents a picked event and each red dot an event not picked at the station (that occurred during its on-time). From such a raw distribution, we derive the distribution of detection probabilities by discretizing the distances and magnitudes (2 km steps and 0.1 magnitude unit steps in Fig. 2). For each magnitude/distance-node, we sample all information about picks (red and green dots) within a distance of 0.1 magnitude unit, hereby converting relative distances to relative magnitude differences by applying the attenuation relation as used by the network. The resulting detection-probability distribution shows the overall trend of increasing probability with increasing magnitude and decreasing probability with decreasing distance (Fig. 2 right). Between magnitudes 3 and 3.5 one can see low detection probabilities for short distances that do not match with the overall trend. Such artifacts are due to very few picks samples and therefore we apply a smoothing procedure to these distributions. With increasing magnitudes and decreasing distances, detection probabilities cannot decrease (Fig. 2 top).

**Fig. 2**: *We derive station detection capabilities probabilities for each station of the network from pick information stored in the catalog. (Left) For each event, we analyze whether or not the station (blue triangle) has picked (recorded) this event during its on-time. (Bottom) Picked and missed events are plotted in green and red, respectively, given the event's magnitude and distance of its hypocenter to the station. (Right) Distribution of detection probabilities derived from sampling the pick information at magnitude/distance-nodes and computing the probability as the number of picked events over the total number of events that occurred during the station's on-time. (Top) Smoothed distribution of detection probabilities.*

Fig. 3 shows how well the derived detection probabilities match the signal strength at the seismometers. The M_{L}=1.3 event (red star) has a hypocentral depths of approx. 15km and thus the distances to station POB and PSP are very similar. However, the signal strength varies strongly due to the different types of stations: POB is a borehole station while PSP is located on sediments. The signal at POB is very pronounced and the respective detection probability (white square) is very high. At PSP the signal can barely be seen and the detection probability for such an event is approx. 0.5.

**Fig. 3**:*Signal strengths of waveform recordings match the computed detection probabilities. The insets show the detection-probability distributions of the four stations (yellow triangles) on the map. The detection probability for the marked M _{L}=1.3 event (red star) is marked as a white square in each distribution. Overlayed is the seismogram of the event as it was recorded at the respective stations.*

## Probabilities of Detection

The probabilities of detection of events not only depend on the station capabilities but also on the procedure the network employs for detecting events. The main parameter is the number of stations that need to detect a signal before the location procedure is started, the so-called triggering condition. Usually networks use a triggering condition of four stations before attempting to locate an event. This means that if an earthquakes creates a signal only at three stations, such an event will remain undetected as no location procedure will be started. Additionally, one needs to know which stations are part of the triggering system. Often networks use data from foreign stations to better constrain the location of events in the border regions, however, these stations are not used in the triggering. The events needs to be detected first and then additional data is pulled for a better location.

To compute the probability of detection of an event with given magnitude at a given location, we combine, according to the triggering condition, all detection probabilities of all active stations of the triggering system. Given a triggering condition of four stations, we compute the probability that at least four stations of the aforementioned set of stations have detected such an event at the given location as shown in Fig. 4.

**Fig. 4**:*Probability of detection for earthquakes of magnitude M=2.5 on 1 January 2008. For most of the mainland Italy, the detection probability reaches 0.999 as indicated by the inner contour line.*

These detection probabilities can be computed for any point in time and they reflect the detection capabilities of the set of station that are in operation at that point in time. Every change in station activity may lead to a change in detection probabilities. This is in particular true for detection probabilities of lower magnitudes as can bee seen in this animation of daily changes in detection probabilities for earthquake of magnitude M=1.5 in Italy (All stations active at each day are indicated by gray triangles).

## Probability-based Magnitude of Completeness

Detection probabilities can be computed for each magnitude level. To derive the completeness value, one needs to define the target probability level, e.g. P_{T}=0.999, at which recordings should be considered complete. Then, for each node on the map, the completeness magnitude, M_{P}, is the lowest magnitude for which the detection probability is greater or equal the target completeness level, see Fig. 5 as example.

**Fig. 5**:*Probability-based magnitude of completeness on 1 January 2008 at the P _{T}=0.999 level. All mainland of Italy is covered at the completeness level of M_{P}=2.9.*

For further details, please see our publications.