*6.1.3 Hopkins*

In 1848, William Hopkins [14] identified two types of seismic waves traveling with different velocities. He pointed out the relation between the corresponding arrival time difference and the distance from the point of observation to the origin of seismic waves. At that time the great advantage of using time differences derived from single recordings was the insensitivity to clock reading errors. The definition of the time difference Tps Eq. 8 and the station delays SDps Eq. 9 implies that (Tps - SDpsi) is zero at the origin. Hence, we define the following cost function:

$$\text{cost function} = \text{MEAN}(\text{ABS}(\text{Tps}\_i - \text{SDps}\_i - \text{r}\_i/\text{Vps})) \tag{12}$$

### **6.2 Cell-hit methods**

The cell-hit methods apply to the various graphical location methods. PGV or arrival times observed at single or couples of seismic stations restrict the possible hypocenter locations to surfaces within the 3D volume. Each surface may hit a distinct assembly of cells of the 3D grid. The number of hits are added up for each cell and the variety of 3D surfaces given by seismic data. The cell with the maximum hit count is taken as the hypocenter location.

Graphical location methods are based on the simple PGV or travel-time distance relations we derived in the previous section. In case the observed data are exactly equal to data calculated by these relations, all surfaces would have one common point and the hit count of the corresponding cell would be the number of surfaces. In fact, the simple relations cannot exactly predict the observed data and these data may also include observational errors. We consider these uncertainties by weighting cell-hits according to their proximity to the surfaces containing the hypocenter. Instead of rating cell-hits as either 1 or 0 we define the following Gaussian weighting schema.

$$\text{hitweight} = \exp\left(-\text{CD}^2/\left(2 \ast \text{sign}\text{m}^2\right)\right) \tag{13}$$

CD ¼ Lij � Dij,klm (15)

CD ¼ Ri � Di,klm (16)

Lij = (Tpi -Tpj) \* Vp...difference of the travel path lengths of stations i and j to

*Seismological Data Acquisition and Analysis within the Scope of Citizen Science*

Dij,klm ...difference of the distances from the center of grid cell [k,l,m] to stations

The PS-circle method is the graphical complement to the Hopkins cost function method. The distance Ri of station i to the hypocenter follows from Eq. 8. The radius Ri defines a circle, in 3D a sphere, which describes the geometrical loci of possible

In 3D Apollonius and PS-circles expand to spheres and hyperbolas to hyperboloids. Spheres with radii smaller than the hypocenter depth do not reach that depth level, and hyperboloids with high eccentricity may not intersect within the grid at greater depth levels. We normalize the cell-hit counts by the sum of cell-hits at each

The location methods presented in the previous subsection work without interaction of the user. Numeric output comprises the hypocenter coordinates (longitude, latitude, focal depth) and in case of amplitude based methods (Kanamori, Apollonius) the magnitude MSS\_M. Focal time, which could be an output of the Geiger and Hyperbola methods, is not documented because we use the trigger time for event identification. However, it would be desirable to get information about the uniqueness and accuracy of the solution. Professionally used location methods like 'HYPOELLIPSE' [16] or 'NonLinLoc' [17] derive confidence ellipsoids from the analysis of the linearization applied to solve the non-linear location problem or by using the probability density function. Here, we content ourselves with visual

The question is, how to visualize the 3D volume of the cost-function or the cellhit values in order to identify the global extremum and to value it against competing local extremes. We chose presentation of the grid values in the different search grid depth levels. First, we show these graphics with synthetic data for an earthquake in the center of the MSS network and at focal depth of 8 km. We take the Kanamori method as an example for the cost function methods (**Figure 10a**) and the Hyperbola method as a representative of the cell-hit method (**Figure 10b**). The cost function low and the maximum cell-hit count areas are clearly und uniquely confined. The visual identification of the optimum hypocenter depth is possible with an accuracy of +/� 1 km. Of course, presuming synthetic PGV, Tp, Tps data and correct program codes presumed, all six methods find the correct hypocentre. Next, we test all six methods with PGV, Tp, and Tps data of the ML = 2.5 earthquake on 14th June 2020 in the center of the MSS-network. **Figure 11** shows

hypocenters. The corresponding proximity length CD is:

Ri = (Tpsi – SDpsi) \* Vps...radius of the PS-circle

*6.2.4 Additional comment to cell-hit methods*

*DOI: http://dx.doi.org/10.5772/intechopen.95273*

depth level to account for this characteristic.

**6.3 Visual check of the location quality**

checks of the location quality.

**49**

Di,klm.....distance of station i to the center of the grid cell [k,l,m].

hypocenter

*6.2.3 PS-circle*

i and j.

Both, CD and sigma have the dimension of a length. CD quantifies the proximity of the center of the grid cell [k, l, m] to the surface and will be specified for each cell-hit method. The parameter sigma considers the data uncertainty.

#### *6.2.1 Apollonius-circle*

Given PGV-values at two stations, these values determine the ratio of the distances from these two stations to the hypocenter according to Eq. 1. The geometrical loci of the hypocenters that fulfill this condition are Apollonius circles in 2D or spheres in 3D [10]. The examination of the accuracy of the Apollonius circle method suggests combining only stations with relative high PGV values with stations of similar or lower PGV values. The combination of low PGV stations does not contribute to an accurate localization.

The proximity length CD of the Apollonius circle method is:

$$\text{CD} = \text{R}\_{\text{p}} - \text{D}\_{\text{p,klm}} \tag{14}$$

Rp … radius of index "p" Apollonius circle

Dp,klm … distance of the center of grid cell [k, l, m] to the Apollonius circle center Cp

The formulae for how to calculate the Apollonius radii Rp and the coordinates of the circle centers Cp are given in the Appendix.

#### *6.2.2 Hyperbola*

Next, we consider the application of the hyperbola method to P-wave travel times Tp. This method dates back to the work of Andrija Mohorovičić ([15]) during his analysis of the Kupa earthquake und the detection of the crust–mantle boundary, the Moho. The travel time differences (Tpi -Tpj) between the stations i and j define Lij, the difference in length of the ray paths from these stations to the hypocenter. Assuming, that Vp = constant, the geometrical loci of the hypocenters are hyperbolas in 2D, or hyperboloids in 3D. The proximity length CD for the hyperbola method is:

*Seismological Data Acquisition and Analysis within the Scope of Citizen Science DOI: http://dx.doi.org/10.5772/intechopen.95273*

$$\text{CD} = \text{L}\_{\text{ij}} - \text{D}\_{\text{ij,klm}} \tag{15}$$

Lij = (Tpi -Tpj) \* Vp...difference of the travel path lengths of stations i and j to hypocenter

Dij,klm ...difference of the distances from the center of grid cell [k,l,m] to stations i and j.

### *6.2.3 PS-circle*

**6.2 Cell-hit methods**

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weighting schema.

*6.2.1 Apollonius-circle*

center Cp

*6.2.2 Hyperbola*

hyperbola method is:

**48**

contribute to an accurate localization.

Rp … radius of index "p" Apollonius circle

the circle centers Cp are given in the Appendix.

The cell-hit methods apply to the various graphical location methods. PGV or arrival times observed at single or couples of seismic stations restrict the possible hypocenter locations to surfaces within the 3D volume. Each surface may hit a distinct assembly of cells of the 3D grid. The number of hits are added up for each

Graphical location methods are based on the simple PGV or travel-time distance relations we derived in the previous section. In case the observed data are exactly equal to data calculated by these relations, all surfaces would have one common point and the hit count of the corresponding cell would be the number of surfaces. In fact, the simple relations cannot exactly predict the observed data and these data may also include observational errors. We consider these uncertainties by weighting cell-hits according to their proximity to the surfaces containing the hypocenter. Instead of rating cell-hits as either 1 or 0 we define the following Gaussian

Both, CD and sigma have the dimension of a length. CD quantifies the proximity of the center of the grid cell [k, l, m] to the surface and will be specified for each

Given PGV-values at two stations, these values determine the ratio of the distances from these two stations to the hypocenter according to Eq. 1. The geometrical loci of the hypocenters that fulfill this condition are Apollonius circles in 2D or spheres in 3D [10]. The examination of the accuracy of the Apollonius circle method suggests combining only stations with relative high PGV values with stations of similar or lower PGV values. The combination of low PGV stations does not

Dp,klm … distance of the center of grid cell [k, l, m] to the Apollonius circle

Next, we consider the application of the hyperbola method to P-wave travel times Tp. This method dates back to the work of Andrija Mohorovičić ([15]) during his analysis of the Kupa earthquake und the detection of the crust–mantle boundary, the Moho. The travel time differences (Tpi -Tpj) between the stations i and j define Lij, the difference in length of the ray paths from these stations to the hypocenter. Assuming, that Vp = constant, the geometrical loci of the hypocenters are hyperbolas in 2D, or hyperboloids in 3D. The proximity length CD for the

The formulae for how to calculate the Apollonius radii Rp and the coordinates of

*=* 2 ∗ sigma<sup>2</sup> (13)

CD ¼ Rp � Dp,klm (14)

cell and the variety of 3D surfaces given by seismic data. The cell with the

hitweight <sup>¼</sup> exp *:* �CD<sup>2</sup>

cell-hit method. The parameter sigma considers the data uncertainty.

The proximity length CD of the Apollonius circle method is:

maximum hit count is taken as the hypocenter location.

The PS-circle method is the graphical complement to the Hopkins cost function method. The distance Ri of station i to the hypocenter follows from Eq. 8. The radius Ri defines a circle, in 3D a sphere, which describes the geometrical loci of possible hypocenters. The corresponding proximity length CD is:

$$\mathbf{CD} = \mathbf{R}\_{\mathbf{i}} - \mathbf{D}\_{\mathbf{i}, \text{klm}} \tag{16}$$

Ri = (Tpsi – SDpsi) \* Vps...radius of the PS-circle Di,klm.....distance of station i to the center of the grid cell [k,l,m].

#### *6.2.4 Additional comment to cell-hit methods*

In 3D Apollonius and PS-circles expand to spheres and hyperbolas to hyperboloids. Spheres with radii smaller than the hypocenter depth do not reach that depth level, and hyperboloids with high eccentricity may not intersect within the grid at greater depth levels. We normalize the cell-hit counts by the sum of cell-hits at each depth level to account for this characteristic.

#### **6.3 Visual check of the location quality**

The location methods presented in the previous subsection work without interaction of the user. Numeric output comprises the hypocenter coordinates (longitude, latitude, focal depth) and in case of amplitude based methods (Kanamori, Apollonius) the magnitude MSS\_M. Focal time, which could be an output of the Geiger and Hyperbola methods, is not documented because we use the trigger time for event identification. However, it would be desirable to get information about the uniqueness and accuracy of the solution. Professionally used location methods like 'HYPOELLIPSE' [16] or 'NonLinLoc' [17] derive confidence ellipsoids from the analysis of the linearization applied to solve the non-linear location problem or by using the probability density function. Here, we content ourselves with visual checks of the location quality.

The question is, how to visualize the 3D volume of the cost-function or the cellhit values in order to identify the global extremum and to value it against competing local extremes. We chose presentation of the grid values in the different search grid depth levels. First, we show these graphics with synthetic data for an earthquake in the center of the MSS network and at focal depth of 8 km. We take the Kanamori method as an example for the cost function methods (**Figure 10a**) and the Hyperbola method as a representative of the cell-hit method (**Figure 10b**). The cost function low and the maximum cell-hit count areas are clearly und uniquely confined. The visual identification of the optimum hypocenter depth is possible with an accuracy of +/� 1 km. Of course, presuming synthetic PGV, Tp, Tps data and correct program codes presumed, all six methods find the correct hypocentre.

Next, we test all six methods with PGV, Tp, and Tps data of the ML = 2.5 earthquake on 14th June 2020 in the center of the MSS-network. **Figure 11** shows

#### **Figure 10.**

*Depth slices through the search grid visualizing (a), the cost function calculated by the Kanamori method and (b), the cell-hit count calculated by the hyperbola method; bright colors mark the cost function low and the cell hit height.*

#### **Figure 11.**

*Location of the ML = 2.5 earthquake, 14th June 2019 by the (a) Kanamori, (b) Geiger, (c) Hopkins, (d) Apollonius, (e) hyperbola, and (f) PS-circle methods; depth slices through the search grids at the optimum focal depth levels are shown; data points (MSS-stations) and the extent of the search grid are marked; the cost function is shown for (a), (b), and (c), the cell-hit count for (e); Apollonius circles (d) and PS-circles (f) visualize the corresponding focal solutions.*

Apollonius, Geiger, Hyperbola, Hopkins, PS-circle) as described before. Subplots centered up the average of these epicenters show the particular solutions. Furthermore, the bulletin epicenters published by ZAMG (epi\_ZAMG) are included in the subplots. Generally, the four epicenters based on the travel-time data Tp and Tps (Geiger, Hyperbola, Hopkins, PS-circle) cluster together well. We calculate an average of these solutions (epi\_TpTps) and plot it on the map and the subplots. We

*Epicenter solutions of 10 earthquakes in the southern half of the MSS-network: EQ23 is the acronym of the ML = 2.5 earthquake, 14th June 2019, addressed in Figures 3, 4, 5, and 11; triangles mark MSS-stations,*

*Seismological Data Acquisition and Analysis within the Scope of Citizen Science*

*DOI: http://dx.doi.org/10.5772/intechopen.95273*

The epicenter data compiled in **Figure 12** allow for a preliminary assessment of the accuracy of the solutions presented. We take epi\_TpTps as reference und calculate the lateral distances to the four travel-time based epicenter solutions (Geiger, Hyperbola, Hopkins, PS-circle), to epi\_PGV and to epi\_ZAMG. Statistical data

Disregarding outliers, the statistics compiled in **Table 1** indicates that the accu-

Next, we consider the focal depth solutions for ten selected events. **Figure 13** shows the individual solutions gained by the six methods (Kanamori, Apollonius, Geiger, Hyperbola, Hopkins, PS-circle), the mean value of the travel-time based methods MSS\_TpTps, and the bulletin focal depth values from ZAMG. The bulletin solution fits to MSS\_TpTps for seven earthquakes in the Vienna Basin near the VBTF (Vienna BasinTransfer Fault) within the 1 km vertical spacing of the search grid. Foci at depths more than 3 km deeper than MSS\_TpTps are indicated by the

racy of epi\_TpTps (mean of epi\_Geiger, epi\_Hyperbola, epi\_Hopkins, epi\_PScircle) mimics the spacing of the search grid spacing (0.5 km). The accuracy of the bulletin solution (epi\_ZAMG) corresponds to the limitation to two decimals of

also calculate the mean of the two epicenter solutions based on PGV data

(Kanamori, Apollonius) and term it epi\_PGV.

**Figure 12.**

**51**

*gridline spacing of insert plots is 2 km.*

about these differences are compiled in **Table 1**.

longitude and latitude [0.01°] in the report.

depth slices of the search grid at each optimum depth level for all methods. The visualization of the density of cell hits by Apollonius circles and PS-circles was applied for the corresponding methods. The hypocenter solutions differ not only between the methods based on the different data type, but also between the costfunction and cell-hit methods using the same data. The latter discrepancy is due to the different weighting of data by the cost function and cell-hit methods. Therefore the variance of the hypocenter solutions obtained by minimum-cost and cell-hit methods using the same data type may be an indicator of the accuracy or significance of the focal solution. The epicenter localizations of the sample earthquake by the six methods scatter within a circle with a radius of 1.6 km. The focal depth varies between 6 km and 8 km.
