**8. Error estimates in the relative locations of epicenters**

When the azimuthal station coverage of an event is not good, the calculated location can move away from the true location because of the inaccuracy in the used velocity model [27]. **Figure 1** shows that the station coverage was not good, and the shifts were unavoidable in the epicenters we obtained. In this section, we examine the uncertainties (errors) in the relative locations between one epicenter and another. If the errors are small, the patterns of the epicentral distribution are reliable.

Errors in locations could be caused by inaccuracies in the used crustal model. To examine if the errors in the relative locations between two epicenters in a group were small, a location test for the ME and one SE was performed. We chose two aftershocks (the ME, No. 36, and an SE, No. 44, in **Table 1**; both have clear onsets, **Figure 5**). **Figure 10** shows the obtained epicenters of the two aftershocks using the same two data sets of arrival time readings but different crustal models and crustal thicknesses. The solid circle with mb 5.7 shows the location of the mainshock, determined by Wetmiller et al., [4]; the two solid squares show the locations of the ME and SE, obtained using our crustal model and a crustal thickness of 32 km. The two solid circles labeled 32, 34, 36, and 38 mark the epicenters of the same two aftershocks obtained using the GSC crustal model (Vp = 6.2 km/s) and crustal thicknesses of 32, 34, 36, and 38 km, respectively.

#### **Figure 10.**

*The epicenters located using different crustal models for the same 2 aftershocks. The solid circle with mb 5.7 represents the epicenter of the mainshock, determined by Wetmiller et al., [4]; the two solid squares with ME and SE represent the located epicenters using our crustal model with thickness of 32 km. The 2 solid circles side by side show the same two aftershocks, located with the GSC crustal model (Vp = 6.2 km) and crustal thicknesses of 32, 34, 36, and 38 km, respectively. When the crustal thickness was changed, the 2 epicenters moved, but their relative positions were visually unchanged.*

As shown in **Figure 10**, the epicenters of the two aftershocks mainly shifted southwards, and the relative locations of the two aftershocks were visually unchanged.

When the two epicenters were over plotted, only subtle changes in relative distances and relative azimuths between the two epicenters could be found. The absolute locations of two epicenters were determined using the calculated travel times, the same crustal model, and the same two sets of observed arrival times. When the parameters in the crustal model change, an increase or decrease in the calculated travel times causes the epicenter to move accordingly. Since the relative locations are mainly determined by the differences in the two sets of observed travel times (**Figure 5**), when the observed travel time readings were not changed, the relative distances and relative azimuths between the two epicenters could only have subtle changes, due to the change in used crustal model. Therefore, this test shows that errors in a crustal model cause systematic errors in the epicenters of an earthquake group, the errors in the relative locations in a group are very small.

Qualitatively speaking, two major types of errors -- errors in the crustal model and in the phase arrival time readings, cause the errors in the epicenters. The errors in the epicenters caused by arrival time reading errors may be roughly estimated. Along the top trace in **Figure 5**, the Sg – Pg time is δ *t* = 6.97–4.00 = 2.97 s. If the P-wave traveled to station KLN with Vp = 6.2 km/s, and Vp/Vs = 1.74, the distance between the station and the epicenter is ∆ = δ *t* × −= *Vp*/ 1.74 1 ( ) 24.88 km. Since the precision of arrival time readings is to 2 decimal places, the reading error in δ *t* is ±0.02 s, and the error in the station distance is δΔ = ±0.02× − *Vp*/ 1.74 1 ( ) = ±0.17 km. If the error in the crustal model causes a +3.5 km error in the latitude of an aftershock, the total error (caused by the error in the crustal model and the error in arrival time readings) is +3.5 ±0.17 = +3.67 or +3.33 km.

Since the same crustal model and arrival time readings of the common phases at the common stations were used to locate the aftershocks, the signs for the absolute errors in the output files should be same. For example, the epicenter of aftershock *a* is (46.977° ±3.2 km, −66.612 ±3.1 km; #36 in **Table 1**), and the epicenter of aftershock *b* is (46.982° ±3.3 km, −66.613 ±3.2 km; #37). For aftershock *a*, if we take (46.977° +3.2 km, −66.612-3.1 km), then *b* is (46.982° +3.3 km, −66.613-3.2 km). In other words, for all aftershocks, the same sign of the errors (+ or -) needs to be assigned because the same crustal model was used. The error caused by the crustal model dominates the total error in an epicenter in the output files of the location program.

The errors in the relative locations of two adjacent aftershocks may be mathematically estimated using the absolute errors in their epicenters. Assume that the epicenters of any two adjacent earthquakes *a* and *b* are (latitude\_a, longitude\_a) with errors (err\_na, err\_ea) and (latitude\_b, longitude\_b) with errors (err\_nb, err\_eb). If vectors **A** = **A0** + **∆A** = (latitude\_a, longitude\_a) + (err\_na, err\_ea) and **B = B0** + **∆B** = (latitude\_b, longitude\_b) + (err\_nb, err\_eb), then the vector difference is **C** = **B** - **A** = (**B0** – **A0**) + (**∆B** – **∆A**) = **C0** + **∆C**. Then we obtain **C0** = (latitude\_b - latitude\_a, longitude\_b - longitude\_a) and **∆C =** (err\_nb – err\_na, err\_eb - err\_ea). Vector **C0** shows the location of *b* relative to that of *a*, while vector **∆C** shows the errors in the location of *b* relative to *a*. Assume aftershock No. 8 in **Table 1** is *a* and one of its adjacent aftershocks, No. 14, is *b*, the errors in the location of No. 14 relative to that of No. 8 are −0.1 km (2.5–2.6) in latitude and − 0.1 km (1.8–1.9) in longitude. No. 13 and No. 15 are adjacent, and the errors in their relative location are (−0.1 km, −0.1 km).

*Locations of the 1982 Miramichi (Canada) Aftershocks: Implication of Two Rupture Regions… DOI: http://dx.doi.org/10.5772/intechopen.108195*

For any given aftershock in **Table 1**, its neighbor events can be found by comparing its distances to other aftershocks. For a given aftershock the distance from its closest neighbor event, the module of its **∆C** with its closest neighbor, and the ratio of this module over this distance are listed in **Table 1**. The average of the modules is 0.183 km, which is much smaller than the gap indicated by the line with an arrow at Az 38° in **Figure 2** or that indicated by the vertical line in **Figure 7**; therefore, the pattern of the obtained hypocenters is reliable.
