**5. Possible errors in the modeled focal depth**

We used the differences in arrival times between synthetic regional depth phases and their reference phases to measure focal depth from the observations. The *P*- and *S*-wave velocities in the crustal model determine the arrival times of these phases. When we generate synthetics we also need the focal mechanism and the earthquake location, but the errors generated by these two factors are negligible.

#### **5.1 The error in the modeled focal depth caused by the crustal model**

Travel times of regional depth phases and their reference phases are determined by the crustal structures through which the phases propagate. As such, most of the error in modeled focal depths comes from the crustal velocity model used.

#### **(A) The error caused by the velocity model**

To evaluate errors arising from velocity uncertainty in the crustal model, we generated synthetic seismograms (at an epicentral distance of 2.16º) using our default crustal model given in Table 1 (model 1), with focal depths from 2 to 23 km. We then reduced the *P*- and *S*wave velocities of the crustal model by 10% and generated another set of synthetics with the same distance and depths. Fig. 18 (Ma and Eaton, 2011) shows the differential times between *sPmP* and *PmP* phases for these two models. The time delay obtained by subtracting these differential times is approximately linear with focal depth. Focal depths, estimated using the RDPM method by treating one set of synthetic traces as observed seismograms, differ by 9.5 – 12% (Fig. 18, bottom). The differences are on the order of 11%, slightly greater than the 10% change in velocity. These numerical tests indicate that the level of uncertainty in the velocity model propagates, at approximately the same order of magnitude, into focal-depth uncertainty.

Fig. 17. *sPmP* and *PmP* modeling for two aftershocks of the 2003/12/22 Central California *M*W 6.5 earthquake. Trace PAS/BHZ/03Dec23 18:15:08.6 is the record of the aftershock 2003/12/23/ 18:17:11.0 *M* 4.9 at station PAS. Trace PAS/BHZ/03Dec23 05:27:41.0 is the record of the aftershock 2003/12/23/ 05:30:19.0 *M* 4.5. Traces 060 and 065 are the synthetic waveforms generated with depths 6 and 6.5 km at the same station (320 km). The focal

We used the differences in arrival times between synthetic regional depth phases and their reference phases to measure focal depth from the observations. The *P*- and *S*-wave velocities in the crustal model determine the arrival times of these phases. When we generate synthetics we also need the focal mechanism and the earthquake location, but the errors

Travel times of regional depth phases and their reference phases are determined by the crustal structures through which the phases propagate. As such, most of the error in

To evaluate errors arising from velocity uncertainty in the crustal model, we generated synthetic seismograms (at an epicentral distance of 2.16º) using our default crustal model given in Table 1 (model 1), with focal depths from 2 to 23 km. We then reduced the *P*- and *S*wave velocities of the crustal model by 10% and generated another set of synthetics with the same distance and depths. Fig. 18 (Ma and Eaton, 2011) shows the differential times between *sPmP* and *PmP* phases for these two models. The time delay obtained by subtracting these differential times is approximately linear with focal depth. Focal depths, estimated using the RDPM method by treating one set of synthetic traces as observed seismograms, differ by 9.5 – 12% (Fig. 18, bottom). The differences are on the order of 11%, slightly greater than the 10% change in velocity. These numerical tests indicate that the level of uncertainty in the velocity model propagates, at approximately the same order of magnitude, into focal-depth

depth solution for the 18:17 event is 6 km and for the 05:30 event is 6.5 km.

**5.1 The error in the modeled focal depth caused by the crustal model** 

modeled focal depths comes from the crustal velocity model used.

**5. Possible errors in the modeled focal depth** 

generated by these two factors are negligible.

**(A) The error caused by the velocity model** 

uncertainty.

Fig. 18. Errors in the modeled focal depth caused by errors in the crustal model. The upper panel shows the differential times between *sPmP* and *PmP* generated by the default crustal model and by a low-velocity model (90% of the default crustal model). The intersection points between the vertical faint lines and the depth axis are focal depth solutions obtained by the two tiled lines from the same differential time *sPmP*-*PmP* (the height of the faint line bar). The difference between the two solutions is the absolute error. The middle panel shows how the absolute errors change with focal depth. The bottom panel shows the relative errors.

#### **(B) The error caused by the Vp/Vs ratio**

We assumed that the crustal media are Poisson type in which Vp/Vs is 1.732. To examine the possible error in our modeled focal depth caused by the Poisson assumption, we made the following tests: (1) We made one new ratio by adding 5% to 1.732 and used the ratio and the Vp values in our default crustal model to create one crustal model *M*1. (2) We subtracted 5% from 1.732 to form a second new ratio and used this ratio to create crustal model *M*2. We compared the synthetics generated using these two crustal models and the default model and found that the time differences *sPmP*–*PmP* on traces generated with *M*1 and depth 11.2 km, generated with *M*2 and depth 12.9 km, and generated with the default crustal model and depth 12 km are approximately equal. This shows that when the crustal medium differs from the Poisson medium by 5%, the relative error in modeled focal depth is less than 8%.

#### **(C) The error caused by strong interfaces in the crust**

Our default crustal model assumes five layers. The thickness of the fourth layer is 6 km (Model 1 in Table 1). We divided the layer into two parts of equal thickness, keeping the original velocities in the upper layer, but changing the velocities in the lower layer to those of the third layer. We generated synthetics with this new crustal model and the default focal mechanism at distance 2.13° (Fig. 19). The time difference *sPmP*–*PmP* does not change noticeably (on traces 120 and STD), but the shape of "*sPmP*" broadened. This change shows that the *sPmP* phase is not a simple phase; it has the "*sPmP*" from the interface where Vp = 6.6 and 7.1 km/sec; Vs = 3.81 and 4.1 km/sec in the new crustal model. This change demonstrates that if there are strong interfaces above the Moho, the *sPmP* phase can be complex, and can cause time-reading errors.

Fig. 19. Synthetic waveforms generated with the default crustal model and a new crustal model that contains a weak lower velocity layer. Trace 110, 115, 120, 125, 130, and 135 were generated with the new model and depth 11, 11.5, 12, 12.5, 13, and 13.5 km. Trace STD was generated with the default crustal model and depth 12 km.

the following tests: (1) We made one new ratio by adding 5% to 1.732 and used the ratio and the Vp values in our default crustal model to create one crustal model *M*1. (2) We subtracted 5% from 1.732 to form a second new ratio and used this ratio to create crustal model *M*2. We compared the synthetics generated using these two crustal models and the default model and found that the time differences *sPmP*–*PmP* on traces generated with *M*1 and depth 11.2 km, generated with *M*2 and depth 12.9 km, and generated with the default crustal model and depth 12 km are approximately equal. This shows that when the crustal medium differs from the Poisson medium by 5%, the relative error in modeled focal depth

Our default crustal model assumes five layers. The thickness of the fourth layer is 6 km (Model 1 in Table 1). We divided the layer into two parts of equal thickness, keeping the original velocities in the upper layer, but changing the velocities in the lower layer to those of the third layer. We generated synthetics with this new crustal model and the default focal mechanism at distance 2.13° (Fig. 19). The time difference *sPmP*–*PmP* does not change noticeably (on traces 120 and STD), but the shape of "*sPmP*" broadened. This change shows that the *sPmP* phase is not a simple phase; it has the "*sPmP*" from the interface where Vp = 6.6 and 7.1 km/sec; Vs = 3.81 and 4.1 km/sec in the new crustal model. This change demonstrates that if there are strong interfaces above the Moho, the *sPmP* phase can be

Fig. 19. Synthetic waveforms generated with the default crustal model and a new crustal model that contains a weak lower velocity layer. Trace 110, 115, 120, 125, 130, and 135 were generated with the new model and depth 11, 11.5, 12, 12.5, 13, and 13.5 km. Trace STD was

generated with the default crustal model and depth 12 km.

is less than 8%.

**(C) The error caused by strong interfaces in the crust** 

complex, and can cause time-reading errors.


Table 2. New crustal models generated by dividing the first layer in the default model (Table 1) into two parts, making the first part 1-km thick, and changing the *P*- and *S*-wave velocities in that layer in steps. *h* = layer thickness (km); *Vp* = velocity of the *P*-wave (km/sec); *Vs* = velocity of the *S*-wave (km/sec); *ρ* = crustal density (g/cm3).

#### **5.2 The error caused by an error in earthquake location**

To estimate the error in the modeled focal depth caused by the error in earthquake location we can observe Fig. 4 or 5 (or Uski *et al*., 2003; their Fig. 2). In the distance window of 1.8º to 3.0º, when the distance changes, for example, 0.1º (~11 km) at distance 2.2º, the time difference *sPmP*–*PmP* is almost constant. This means that when the earthquake location has an 11-km error in the above distance window, the error in the modeled focal depth caused by the error in earthquake location is negligible.
