Focal Depth Determination for Moderate and Small Earthquakes by Modeling Regional Depth Phases sPg, sPmP, and sPn

Earthquake focal depth is a critical parameter for seismological research, seismotectonic study, seismic hazard assessment, and event discrimination. For most earthquakes with MW ≥4.5, the focal depth can be estimated from the arrival times of the teleseismic depth phase sP (or pP) and its reference phase P. Many seismologists have studied how to detect and use teleseismic depth phases to estimate focal depth (e.g., Goldstein and Dodge, 1999). For smaller earthquakes, focal depths can be estimated jointly while being located with the arrival times of the Pg and Sg phases recorded at close stations. Because stations in a regional network are generally not dense enough to control focal depth, operators often use default focal depths for regional events. If regional depth phases can be identified, an alternative solution for moderate and small earthquakes is to use regional depth phases to estimate focal depth. The P portion of regional waveform records contains three major parts: (1) the P-wave travels directly to the station; (2) the Por S-wave travels upward to the surface in the source region, is reflected or converted at the surface and then travels downward to the Moho (or interfaces), is reflected or refracted there, and then travels upward to the station; and (3) the P-wave travels downward to the Moho (or interfaces), is reflected there and then travels upward to the station. One feature of Pand S-waves is that the amplitude of the S-wave radiated from the source is generally stronger than that of the P-wave by about five times (Aki and Richards, 1980) and the period of the S-wave is longer than that of the P-wave on the same record. From this analysis we know that there are regional depth phases in the P portion of the record and the usable regional depth phases are (1) sPg (the S-wave travels upward to the surface, is converted to a P-wave at the critical angle, then the P-wave travels along or close the surface to the station), (2) sPmP (the S-wave travels upward to the surface, is converted to a P-wave, then the P-wave travels downward to the Moho, is reflected there and travels upward to the station; Langston et al., 2003), and (3) sPn (the S-wave travels upward to the surface, is converted to a P-wave, then the P-wave travels along the Pn path to the station;

. Sketch figures for regional depth phase sPg (upper panel), sPmP (middle), and sPn (bottom). Regional depth phases (sPg, sPmP, and sPn) can be used to estimate focal depth if they and their reference phases (Pg, PmP, and Pn) can be correctly identified. Following Langston (1987) and Bock et al. (1996), we developed a method to use the regional depth phases to determine focal depths. The principle is: (1) calculate synthetics with the reflectivity method (Randall, 1994) at a station with a reasonable range of depths; (2) compare the synthetics with the observed values at the same station; and (3) take as the focal depth of the earthquake the depth at which the synthetic and the observation have similar time differentials (regional depth phase to its reference phase). We previously reported some aspects of the regional depth-phase modeling (RDPM) method (e.g., Ma et al., 2003;Ma and Atkinson, 2006). Here we introduce the RDPM method more systematically and describe the principles and features of the three depth phases in detail. We have proved that the assumptions of depth phase sPmP and its reference phase PmP are correct, and by conducting several tests, found in which regions the regional depth phases are developed and in which they are not, and which factors contribute to errors in the modeled focal depths. We also found that the contents of PmP and sPmP come from different interfaces beneath the source. These findings are useful for researchers who want to use the RDPM method, and especially for identifying the regional depth phases and their reference phases. Because we use regional synthetics as a "ruler" to measure focal depth from observed waveforms, we first describe how to generate synthetics that are suitable to be used as the "ruler" and discuss some features of the regional depth phases.
www.intechopen.com Focal Depth Determination for Moderate and Small Earthquakes by Modeling Regional Depth Phases sPg, sPmP, and sPn 145 2. Synthetic regional depth phases sPg, sPmP, and sPn To generate synthetics we need a crustal model, earthquake location, focal mechanism, and focal depth. To generate synthetics for smaller earthquakes, the source time function is not important. We use a triangle as the source time function. Because the focal mechanism does not determine the arrival times of the seismic phases-the crustal structures determine the arrival times-the crustal model is a key factor in generating synthetic regional depth phases. Western Quebec is one of the more active seismic zones in eastern North America and the crustal structures are relatively well known. Mereu et al. (1986) "conducted a major long-range seismic refraction and wide-angle reflection experiment across the Grenville province of Canadian Shield," and obtained some crustal models. After studying these crustal models and modifying them slightly, we obtained one crustal model and put it in our program package as the default crustal model ( Fig. 2; model 1 in Table 1). Because the focal mechanisms in western Quebec are predominantly thrust type (e.g., Adams et al., 1989;Bent and Perry, 1999;Ma and Eaton, 2007), we used a thrust type focal mechanism as the default (Fig. 2, bottom left). Because regional depth phases are easier to discern on displacement records than on velocity records, we used displacement records in the RDPM. All the synthetic and observed waveforms in this chapter are the vertical component.   Table 1. The crustal models. Model 1 is the default in the RDPM program package. Crustal models 2, 3, 4, and 5 were formed by deleting the last layer successively from model 1. h = layer thickness (km); Vp = velocity of the P-wave (km/sec); Vs = velocity of the S-wave (km/sec); ρ = crustal density (g/cm 3 ).

Synthetics generated at different distances with a fixed focal depth
To observe features of the regional depth phases that are displayed when the distance changes, we generated synthetic waveforms at distances ranging from 0.3º to 4.8° and plotted them (Figs. 3,4,and 5). Fig. 3 shows that the sPg phase is well developed at distances ranging from 0.7º to 0.9º (trace 070 to 090). The distance range within which sPg is well Fig. 3. Synthetic waveforms generated with depth 12 km, azimuth 236º, at distances 0.3º to 2.0º (also used to generate Figs. 4,5,6,7,and 8). Trace number = distance in degrees  100. Trace 030 was generated at distance 0.3º. Traces are aligned on the first phase. On trace 070 phases Pg and sPg and on trace 140 phases PmP and sPmP are labeled.
www.intechopen.com Focal Depth Determination for Moderate and Small Earthquakes by Modeling Regional Depth Phases sPg, sPmP, and sPn 147 developed changes with focal depth: the range shifts farther as the focal depth increases. The time difference sPg-Pg changes very slightly with distance. For the distance range of about 1.0º to 1.7º (trace 100 to 170), Pg, PmP, sPg, and sPmP co-exist. Fig. 4 shows that Pg disappears at 1.6º (or Pg and PmP merge there; trace 160); sPg disappears at 1.9º (trace 190). For the distance window of about 1.8º to 2.8º (trace 180 to 280), the waveforms are quite simple. The first phase is Pn (generally weak); the second phase is PmP and the third phase is sPmP. Fig. 5 shows that sPn stands out at about 3.0º (trace 300). The time difference sPn-Pn is independent of distance. For distances larger than 2.9º (trace 290), waveforms become complex. At about 4.1º (trace 410) there is another distance window in which waveforms are relatively simple. Fig. 4. Synthetic waveforms generated with depth 12 km, azimuth 236º, at distances 1.0° to 3.0°. Trace 100 was generated at distance 1.0°. Traces are aligned on Pg or PmP. The distance window in which waveforms are simple is from about 200 to 300 km (trace 180 to 280). The Pn phase is weak. Traces 100 to 200 correspond to the early parts of those traces with the same labels in Fig. 3, but with amplitude enlarged and timescale expanded. . Synthetic waveforms generated with depth 12 km, azimuth 236º, at distances 2.0º to 4.8º. Top trace 200 was generated at distance 2.0°. Traces are aligned on Pn. The sPn phase stands out at 3.0º (trace 300), but is buried at closer distances. After 2.8º (trace 280) waveforms become complex. Around trace 410 (4.1°) waveforms are simple again.

Synthetics generated with a range of focal depths at fixed distances
To observe how regional depth phases change with focal depth, we generated synthetic sPg, sPmP, and sPn with a range of depths at fixed distances 0.9º, 2.1º, and 4.1º. Fig. 6 shows that the time difference sPg-Pg becomes progressively larger with focal depth. The position of sPg shifts by about half a cycle when the depth changes by 1 km. This means that the time difference sPg-Pg is very sensitive to focal depth. At distance 0.9°, sPmP is not well developed. Fig. 7 shows that the time difference sPmP-PmP becomes larger as depth increases. The position of sPmP also shifts by about half a cycle when focal depth changes by 1 km. The Pn phase is also a depth phase, but it is not as sensitive as sPmP to focal depth. For example, on trace 210, the time difference between Pn and PmP is about half that www.intechopen.com Focal Depth Determination for Moderate and Small Earthquakes by Modeling Regional Depth Phases sPg, sPmP, and sPn 149 between sPmP and PmP. The time difference Pn-PmP changes obviously with distance (Fig.  5). These features of Pn can be used to identify sPmP in its distance window (200 to 300 km). Fig. 8 shows how the time difference sPn-Pn changes with focal depth. Because the sPn phase is stronger than Pn, it is possible that some of the observed "Pn" phase beyond 300 km is sPn. Fig. 6. Synthetic waveforms generated at distance 0.9º with depths from 1 to 33 km. Trace number = depth in km  10. Trace 010 was generated with depth 1 km at distance 0.9°.

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Earthquake Research and Analysis -Seismology, Seismotectonic and Earthquake Geology 150 Fig. 7. Synthetic waveforms generated at distance 2.1º with depths from 1 to 33 km. Trace 010 was generated with depth 1 km at distance 2.1º. On trace 070 phases PmP and sPmP are labeled. The Pn phase is weak, and is labeled on traces 120, 220, and 330. Fig. 8. Synthetic waveforms generated with depths of 1 to 33 km at distance 4.1°. Trace 010 was generated with depth 1 km at distance 4.1º. The Pn phase is weak; sPn is stronger than Pn. From trace 260 (26 km) sPn merges with other phases. Traces are aligned on the Pn phase.

Synthetics generated with different focal mechanisms
To generate synthetics we need a focal mechanism. Generally, no focal mechanism solutions are available for small earthquakes. To solve t h i s p r o b l e m , w e u s e d a d e f a u l t f o c a l mechanism for all earthquakes to generate synthetics to measure focal depths. Because focal mechanisms do not determine the arrival times of seismic phases, we used a default focal mechanism to generate synthetic phases for their arrival times.

Demonstration for the assumed sPmP and PmP phases
In eastern North America many P portions of waveform records are similar to trace CRLO/EHZ in Fig. 9. On this trace, the first weak phase is Pn. We assumed that the second phase is PmP and the third phase (the largest) is sPmP. To demonstrate that the assumptions are correct, we prepared Fig. 9 using explosive and earthquake source models. On trace EXPL/140, no strong assumed depth phase appears at the position corresponding to that on trace CRLO/140 which was generated with the same depth and crustal model. The reason is that an explosive source does not directly generate S-waves. According to the definition of sPmP, the phase should arrive at a station progressively later as the focal depth increase because the total path length becomes longer. The third phase on traces CRLO/130, 135, 140, 145, and 150 arrives progressively later as the depth increases. This feature and the absence of the strong phase on EXPL/140 indicate that the "assumed sPmP" is sPmP. , an explosive source model (traces EXPL/140 to 150), and the P portion recorded at station CRLO generated by a small earthquake (1995/09/12, m N 3.7, west Quebec; trace CRLO/EHZ). CRLO/130 was generated with depth 13 km at distance 2.10º and azimuth 283°. EXPL/140 was generated with depths 14.0 km. Trace CRLO/EHZ is the P portion generated by the small earthquake. The modeled focal depth for this earthquake is 14.5 km.
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The waveform contents of the sPmP on trace 6/150, for example, contain contributions from all interfaces beneath the source. From the definition of PmP we know that as focal depth increases, PmP should arrive earlier, because the total travel path becomes shorter. Generally, we use the phase recorded in the distance window of about 200 to 300 km. At these distances, the feature that PmP arrives earlier as focal depth increases is not easy to examine without Pg as the reference phase. So we traced the assumed PmP to close distances where Pg exists. Fig. 11 is the synthetic waveforms generated with an explosive source model. Trace 210 in Fig. 11 is similar to trace EXPL/140 in Fig. 9, which was generated with depth 14 km at distance 2.10°. Let us trace the assumed PmP in Fig. 11 from trace 080 to trace 150 where both Pg and the assumed PmP exist. At such a close distance, the first phase is Pg. We then generated synthetics at fixed distance 0.8° with depths of 1.0 to 35.0 km (Moho depth). Fig. 12 shows that the two phases become closer as depth increases, and they merge at the Moho. This test shows that the assumed PmP is PmP. Based on these tests, the assumed phases PmP and sPmP on trace CRLO/EHZ are PmP and sPmP, because these two phases have the same features as those on traces CRLO/130 to CRLO/150.  Table 1 at distance 2.10° and with depths 13, 14, and 15 km. Trace Group 2 (2/130, 2/140, 2/150) was generated with a two-layer crustal model (Model 5 in Table 1); Group 3 with a three-layer crustal model (Model 4 in Table 1); Group 4 with a four-layer crustal model (Model 3 in Table 1); Group 5 with a five-layer crustal model (Model 2 in Table 1); Group 6 with a sixlayer crustal model (Model 1 in Table 1). Fig. 11. Synthetic waveforms generated with depth 13 km, an explosive source model, and crustal model 1 in Table 1, at distances of 0.5° to 3.0°. Trace 210 was generated at 2.10°. The Pg phase disappears at about 1.5° (trace 150). The PmP phase is clear from 1.7° to 2.8° (traces 170 to 280).  Table 1. Trace 130 in this figure is the same as trace 080 in Fig. 11. Because the total path length becomes shorter with focal depth, the PmP phase arrives progressively earlier with depth.

Comparison of RDPM focal depths with those from other methods
In eastern Canada and the New York State region, some moderate and strong earthquakes have reliable focal depth solutions. For the same earthquakes we compared focal depth solutions obtained by RDPM with those obtained by other methods. We found that the consistency is good.

sPn modeling for the M W 5.0 2002/04/20 Au Sable Forks, New York earthquake
We analyzed the waveform records for this earthquake and found many sPn records. We modeled the sPn and Pn phases recorded at a POLARIS station, Canada (Fig. 13). From sPn and Pn paths in a one-layered crustal model of Poisson medium, the following equation calculates focal depth using the time difference sPn-Pn: ; V p1 and V p2 are P-wave velocities in and beneath the crust, respectively; h is the focal depth; and t is the differential time. The differential time is independent of station distance and crustal thickness. Even if we have only one pair of reliable sPn and Pn observations, we can obtain a focal depth solution with a small error without considering the earthquake location error. The time difference sPn-Pn on trace ACTO/HHZ is 4.06 sec. When we take V p1 = 6.25 km/sec and V p2 = 8.0 km/sec, the focal depth from equation (1) is 11.7 km; this depth is close to the focal depth (about 11.5 km) determined by RDPM. Fig. 13. Phase sPn modeling for 2002/04/20 Au Sable Forks, New York, M W 5.0 earthquake. Trace ACTO/100 was generated with depth 10 km at station ACTO (517 km). Trace ACTO/HHZ is the earthquake observation; it matches a trace somewhere between ACTO/110 and ACTO/115. We can take 11 km or 11.5 km as the modeled focal depth. The time difference sPn-Pn on trace ACTO/HHZ is 4.06 sec. Using this number in equation (1) gives a focal depth of 11.7 km.
To compare the RDPM depth, we analyzed the teleseismic depth phase sP and its reference phase P at YKA (Yellowknife Array, Canada). Fig. 14  where V P is the P-wave velocity in a one-layered crustal model of Poisson medium, α is the P-wave take-off angle, t is the differential time, and h is the focal depth. If we take V P = 6.25 km/sec and α = 29.68° the focal depth from equation (2) is 12 km.  (2) is about 12 km.

sPg modeling for the m N 4.3 1993/11/16 Montreal south earthquake and its aftershocks
Many sPg (and sPmP) records are available in eastern Canada and New York region. As an example we selected the records from station MNT (37 km from a main shock and its   Table 2 of Ma, 2010). Fig. 15 shows that the main shock is the deepest and the last aftershock is the shallowest, and the depth difference between, for example, No. 15 and 16, is about 0.3 km.
4.3 sPmP modeling for the M L 5.1 1983/10/07 New York region earthquake and its aftershock (No. 5 in Table 2

of Ma 2010)
We selected the records from station SBQ (244 km) as an example. In Fig. 16, trace SBQ/SHZ 10:18:12.0 is the record of the main shock at station SBQ, and trace SBQ/SHZ 10:39:06.0 is the record of its aftershock. From the time differences sPmP-PmP on the synthetics SBQ/079 and SBQ/085, we found that the aftershock is shallower than the main shock by about 0.5 km. The modeled focal depth for the main shock is 8.5 km. The focal depth obtained by the Geological Survey of Canada is 10.0 km. From the first record we see the Pn phase clearly, but from the second record we cannot. This means that the Pn phase disappeared or was too weak to be measured.  Table 2 of Ma, 2010 and one of its aftershocks (1983/10/07 10:39:39.0, M L 3.5). The top trace is the record of the earthquake at station SBQ; the second trace is the record of the aftershock. Traces SBQ/079 and SBQ/085 are synthetic waveforms generated with depths 7.98 and 8.5 km at station SBQ. The subtle differential times sPmP-PmP show that the two earthquakes occurred at different depths. On the second trace the Pn phase is not measurable.

sPmP modeling for two aftershocks of the M W 6.5 2003/12/22 California earthquake
Many regional records are available for the main shock and its aftershocks. We retrieved the records of some aftershocks at stations within 5º from IRIS for analysis. We found that two aftershocks have clear sPmP phase records at stations VCS (2.76º), CHF (2.89º), PAS (2.87º), and MWC (2.92º). We modeled the sPmP and PmP at station PAS, and obtained focal depths 6.5 km for the 05:30 event and 6 km for the 18:17 event (Fig. 17). The preliminary focal depths obtained by the local network for the two aftershocks are 5.9 km and 6.9 km, respectively. Traces 060 and 065 are the synthetic waveforms generated with depths 6 and 6.5 km at the same station (320 km). The focal depth solution for the 18:17 event is 6 km and for the 05:30 event is 6.5 km.

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.

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 Swave 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.
www.intechopen.com Focal Depth Determination for Moderate and Small Earthquakes by Modeling Regional Depth Phases sPg, sPmP, and sPn 159 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 M1.
(2) We subtracted 5% from 1.732 to form a second new ratio and used this ratio to create crustal model M2. 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 M1 and depth 11.2 km, generated with M2 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.  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/cm 3 ).

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.

The error caused by the focal mechanism
Focal mechanism determines the radiation pattern of seismic waves but it does not determine arrival times of seismic phases (e.g., Ma and Atkinson, 2006;their Fig. 11). However, the focal mechanism partially determines the shapes and amplitudes of waveforms and thus may cause an error in time readings. If the observed PmP and sPmP are clear, but the synthetic PmP or sPmP are not, we can change the station azimuth, generate new synthetics, and then make comparisons again. So, the arrival time reading error caused by the focal mechanism is negligible. To reduce the reading error, we use the arrival times of peaks of sPmP and PmP.

Possible factors that determine the development of the regional depth phases
Many factors determine the development of regional depth phases. The key factors are the crustal structures at the free surface and at the Moho. We tested the effect of crustal structure at the free surface on the development of the regional depth phases. We divided the first layer of the default crustal model into two parts, making the first layer 1-km thick. We changed the velocities in the new first layer gradually to obtain new crustal models (

Outline of the procedure to identify the regional depth phases
There are distance windows in which regional depth phases are developed. Phase sPg is developed well within 100 km, sPmP is developed well within about 200 to 300 km, and sPn is developed at more than 300 km. Figs. 3, 4, and 5 show these distance windows. Because time differences sPg-Pg, sPmP-PmP, and sPn-Pn are not sensitive to station distance, we can align all records generated by an earthquake by station distance to identify these regional depth phases. The procedure has the following steps: 1. Retrieve the catalogue from the official website of GSC (Geological Survey of Canada); 2. Select an earthquake to determine its focal depth using RDPM; 3. Retrieve the pick file for the earthquake; 4. Retrieve all the available waveform records from the same website; 5. Based on the station distances in the pick file, arrange/display the waveform records; 6. In the above distance windows, align the records first and then search the regional depth phase and its reference phase pairs.
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If two pairs (one pair = regional depth phase and its reference phase) on two records at two stations have similar differential times, the pairs may be treated as candidates for modeling. We can model one pair or two pairs using RDPM to obtain the focal depth. The modeling procedure is shown in Fig. 9. If only one record is available at a distance window, the period feature (sPg, sPmP, and sPn have longer periods than their respective reference phases) may be used for the identification. The feature that Pn, PmP, and sPmP have relative positions on the record (see Fig. 7) can also be used to identify sPmP and PmP. Dineva et al. (2007;their Fig. 9) provided an excellent example for regional depth phase identification. If more than one station has regional depth phase records for the same earthquake, the focal depth solutions obtained at different stations should be similar (see Ma and Atkinson, 2006; their Table 1).

Summary and discussion
There are many small earthquakes in eastern North America. These earthquakes do not have measurable teleseismic depth phases, and generally do not have close (<40 km) waveform records. No focal depths can be reliably estimated with either teleseismic depth phases or close seismic signals for most of these earthquakes, but the depths are crucial information for many topics, both theoretical and applied. On regional waveform records, one or more phases are well developed between the first arrivals (Pg or Pn) and the S-wave train, and one or two of them are regional depth phases. Within about 100 km, the sPg phase is well developed on some records of earthquakes as small as m N 1.5. In the distance window from about 200 to 300 km, the sPmP phase is well developed on some records of earthquakes as small as m N 2.0. Beyond 300 km, the sPn phase is developed on some records of moderate and sub-moderate earthquakes. All these regional depth phases can be used to estimate focal depth. When we generate synthetic depth phases, we use a default crustal model and default focal mechanism, and even a default station azimuth and default instrument response. In this way, we can conveniently generate synthetics for any small earthquake, and use the synthetic depth phases to estimate focal depth for that earthquake, if the arrival time difference between one observed depth phase and its reference phase is available. We compared our modeled focal depths with those that were reliably obtained by other methods and found that the consistency in the comparisons is good. The errors in our modeled focal depths are caused mainly by the crustal model used. The relative errors due to crustal model are estimated as within 15% when the error in the crustal velocity model is 10%, or when the thicknesses of layers in the crustal model have some errors, or the crustal medium differs from the assumed Poisson medium. We have analyzed the regional depth phases for many earthquakes (Ma and Atkinson, 2006;Kim et al., 2006;Dineva et al., 2007;Ma and Eaton, 2007;Atkinson et al., 2008). For this chapter we selected some special cases to show that our modeling method is simple, reliable, and suitable for all regions where regional depth phases are developed.

Acknowledgments
This chapter was adapted with permission from the paper "Focal Depth Determination for Moderate and Small Earthquakes by Modeling Regional Depth Phases sPg, sPmP, and sPn" published by Bull. Seism. Soc. Am. It covers the author's long-time research results. Most of