Resolution and Application

Some snapshots of resolution tests and identifications of P$^d$P are discussed in the following section. The complete resolution tests can be found on the CD.

Figure 5.4: Resolution test of sliding-window fk-analysis. A synthetic signal recorded at an array with YKA configuration is overlayed with white noise. 11 damped sine signals with signal lengths of 2 s and varying amplitudes are contained. The waves arrive as a coherent wave with a slowness of 7.75 s/$^{\circ}$ and a backazimuth of 225$^{\circ}$ at the array. These values are marked by a diamond. The amplitudes of the signals are set relative to the noise amplitude starting at 100% (i.e. signal to noise ratio 1:1) to 8.3% (i.e. signal to noise ratio $\sim$ 1:12). The maximum of the fk-diagram is marked by a circle. a) Singel trace of the array recording. The travel times of the sinusoidal signals, which are buried by the noise, are marked by the boxes. 6 fk-diagrams for different time windows are displayed in b) - g).
b) Signal amplitude 100% of the noise amplitude. The slowness and backazimuth of the wave is well resolved and the coherency of the signal is very good.
c) Signal amplitude 83.3%. Good u and $\Theta$ determination. Coherent signal.
d) Signal amplitude 75%. Good u and $\Theta$ determination. Coherent signal.
e) Signal amplitude 50%. Small slowness and backazimuth deviations.
f) Signal amplitude 41.6%. The signal can still be identified as a coherent signal arriving with the defined slowness and backazimuth.
g) Signal amplitude 33.3%. No signal detectable.

Figure 5.4 shows a resolution test of the sliding-window fk-analysis. For this resolution test, white noise and a damped sinus signal with a dominant period of 1 s and a total length of 2 s are summed. The synthetic signals arrive with the appropriate time shifts for a slowness of 7.75 s/$^{\circ}$ at the 18 stations of the YKA configuration. Figure 5.4a) shows 11 wavelets with a backazimuth of 225$^{\circ}$ and a slowness of 7.75 s/$^{\circ}$ with different amplitudes in one trace. The arrival times of the wavelets, which are totally buried in the noise, are indicated by the vertical boxes. 6 snapshots of the sliding-window fk-analysis are displayed in b) - g). For all sliding-window fk-diagrams the power of the individual panels is normalized to the maximum power of all time windows. The amplitudes of the wavelet relative to the noise amplitude are shown in per cent with decreasing signal amplitudes from b) - g). In Figure 5.4b) the signal and the noise have the same amplitudes (A$_{signal}$/A$_{noise}$ = 1 = 100%). In the fk-diagram the maximum power for the correct slowness (u = 7.75 s/$^{\circ}$, $\Theta$ = 225$^{\circ}$) is indicated by the white circle and the signal is coherent due to the resemblance of the fk-diagram to the ARF. In Figure 5.4c) the amplitude ratio is 83.3%. The colour indicates the lower signal energy due to the normalization of the power to all time windows. Again u and $\Theta$ are correctly determined and the signal is coherent. In Figure 5.4d) and Figure 5.4e) the slowness and backazimuth of the input signals can be resolved and the signals seem to be coherent. Even for an amplitude ratio of 41.6% (Figure 5.4f), the computed slowness and backazimuth are in agreement with the theoretical values of the wavefront and the signal can be interpreted as coherent. Only for an amplitude ratio of 33.3% the determination of u and $\Theta$ fails as the coherent signal is totally covered by incoherent noise.
The assumptions of white noise and a 2 s long sinus signal are artificial. Natural noise and seismic signals show different spectra compared to the synthetic signals used in the resolution test. The small differences along the ray paths to different array stations are the reason that the real signals arriving at the array are not as perfectly coherent as assumed in the synthetic tests. Nevertheless, this resolution test can give estimates on the maximum resolution of the sliding-window fk-analysis. A detection limit of $\sim$50% - 70% is used for the interpretation.

Figure 5.5: Slowness resolution test of the sliding-window fk-analysis. The signal to noise ratio of the coherent signals is 2:1. White noise was added to the traces. All phases arrive with a backazimuth of 225$^{\circ}$. The same damped sinus signal as in Figure 5.4 is used. For each phase the slowness and backazimuth is indicated by the white diamonds. The maximum of the fk-analysis is marked by the white circle. The scale for these fk-diagrams is given in a)
a) Two phases with same backazimuths arrive simultaneously. The slownesses are different: u$_1$ = 7.75 s/$^{\circ}$ and u$_2$ = 1.75 s/$^{\circ}$. In the fk-diagramm two maxima with reduced power can be identified. The slowness and the backazimuth of the signals are well resolved. The gray-scale bar on the right hand side is valid for all fk-diagrams shown here.
b) The two phases with the slownesses u$_1$ = 7.75 s/$^{\circ}$ and u$_2$ = 1.75 arrive with a time difference of 2 s. Phase 1 arrives at 0 s and phase 2 at 2 seconds. Both phases are resolved correctly, although the second maximum is located at 4 s.
c) Three phases with slownesses u$_1$ = 7.75 s/$^{\circ}$, u$_2$ = 4.5 s/$^{\circ}$ and u$_3$ = 1.75 s/$^{\circ}$ arrive with 2 s difference each. The third phase detected in the third panel of c) has only half the amplitude of the other phases and, therefore, has a SNR of 1:1. All three phases are resolved with correct slownesses and backazimuths and at the right time.

The slowness resolution of the sliding-window fk-analysis is another important issue to test. The slowness is the crucial factor to identify P$^d$P phases. It is important to study whether the fk-analysis is able to separate two signals arriving with small time differences and different slownesses.
Figure 5.5 shows three different synthetic tests. As in Figure 5.4, the signals are perfectly coherent and white noise is added to the traces. The signal to noise ratio is 2:1. In Figure 5.5c) the last phase arriving has an amplitude ratio of 1:1. Slownesses and backazimuths of the synthetic signals are marked by white diamonds. The maximum of the fk-diagram is indicated by the white circle. All fk-diagrams are normalized to the maximum value of all diagrams.
Figure 5.5a) shows the fk-diagram of a time window containing two phases with slownesses u$_1$ = 7.75 s/$^{\circ}$ and u$_2$ = 1.75 s/$^{\circ}$. Both phases arrive at the same time at the stations and travel along the same great circle path with a backazimuth of $\Theta$ = 225$^{\circ}$.
The fk-diagram in Figure 5.5a) shows two maxima. These maxima correspond well to the theoretical slowness and backazimuth of the synthetic phases. This test shows that the sliding-window fk-analysis is able to separate these two phases. It is possible to identify the corresponding slowness and backazimuth of each phase on the great circle path arriving simultaneously with different slownesses.
Figure 5.5b) shows three snapshots of the sliding-window fk-analysis with times of 0 s, 2 s, and 4 s. Two phases with 2 s differential travel time (t$_1$ = 0 s and t$_2$ = 2s) and slownesses u$_1$ = 7.75 s/$^{\circ}$ and u$_2$ = 1.75 s/$^{\circ}$ arrive within the 4 s sampled. The first snapshot shows clearly the high slowness of the first phase. The second time window shows a transition between the two phases, when parts of both signals are within the time window of the fk-analysis. The third snapshot shows the slowness of the second phase. The correct slowness of the second phase is resolved too late due to the overlapping of the two phases. Although the second phase is not resolved at the correct time, this test shows that the different slownesses of the two phases can be resolved even if they have only small travel time differences. The difference between the 2 s panel in Figure 5.5b) and the two simultaneous phases in Figure 5.5a) is noticeable. The shifted detection time of phase 2 is taken into consideration for the error analysis.
The slowness difference of the phases studied in the previous test is larger than the difference between P and PP. To test whether the real slowness difference can be distinguished for two phases with small travel time differences, the test in Figure 5.5c) was performed. Three phases with slownesses u$_1$ = 7.75 s/$^{\circ}$, u$_2$ = 4.5 s/$^{\circ}$ and u$_3$ = 1.75 s/$^{\circ}$ arrive with 2 s differential travel time (t$_1$ = 0 s, t$_2$ = 2 s and t$_3$ = 4 s). The first two phases have a signal to noise ratio of 2:1. The third phase has a signal to noise ratio of 1:1 in order to study if a phase with a smaller amplitude can be resolved shortly after a stronger phase. All three phases are correctly determined by the fk-analysis. Even the third phase with the small slowness and the low signal to noise ratio is detected with its correct slowness. This fk-diagram is not as clear as in the other time windows as a result of the lower signal amplitude.
These tests show that the sliding-window fk-analysis is capable of measuring slownesses and backazimuths of phases with different slownesses, even if the phases arrive simultaneously or with a small differential travel time.

Figure 5.6: Backazimuth resolution test for the sliding-window fk-analysis. Two phases arrive with a travel time difference of 2 s with slightly different backazimuths. The first phase always shows a slowness of u = 7.75 s/$^{\circ}$ and a backazimuth of $\Theta$ = 225$^{\circ}$. These values are marked by the white diamond. The second phase shows a backazimuth as marked in the figure ($\Theta$ = 230$^{\circ}$, 235$^{\circ}$ and 245$^{\circ}$). The white point marks the maximum of the fk-analysis. White noise was added to the synthetic signals to produce a signal to noise ratio of 2:1.
a) $\Theta_{2}$ = 230$^{\circ}$. The difference of the fk-diagrams is hardly noticeable and could as well be a result of the different noise conditions in the two time windows.
b) $\Theta_{2}$ = 235$^{\circ}$. The backazimuth deviation of the second phase is clearly resolvable. The maximum of the fk-diagram moves away from the backazimuth of the first phase.
c) $\Theta_{2}$ = 245$^{\circ}$. The difference can be identified without doubt.
A backazimuth difference $\Delta\Theta$ = $\pm$10$^{\circ}$ can be resolved by the sliding-window fk-analysis.

The maximum backazimuth resolution is shown in Figure 5.6. In this test, two phases with different backazimuths arrive with a differential travel time of 2 s. The first phase always travels along a path with a backazimuth of $\Theta$ = 225$^{\circ}$ and a slowness of u = 7.75 s/$^{\circ}$. This phase is marked by a white diamond in all panels. White noise was added to the traces to produce a signal to noise ratio of 2:1.
Figure 5.6a) shows the fk-diagrams of two phases arriving with a backazimuth difference $\Delta\Theta$ =  5$^{\circ}$. A small difference of the two fk-diagrams is detectable, but could also be a result of different noise conditions in the two time windows. An obvious difference of the fk-diagrams is resolvable when $\Delta\Theta$ = 10$^{\circ}$ (Figure 5.6b). The maximum of the fk-diagram is clearly shifted away from the backazimuth of the first phase. When the backazimuth difference is 20$^{\circ}$ as shown in Figure 5.6c) a clear identification is possible. The difference is obvious and the backazimuth is resolved correctly.
These tests demonstrate a backazimuth resolution of at least $\Delta\Theta$ = $\pm$10$^{\circ}$. All greater deviations from the great circle path can be interpreted as asymmetric paths resulting from reflections away from the great circle path or phases arriving from different events or noise sources close to the array producing more or less coherent noise.

Figure 5.7: Sliding window fk-analysis for the event 04-jun-1993 10:49 under North-Halmahera (depth: h = 15 km, distance: $\Delta$ = 98.6$^{\circ}$, $\Theta$ = 295.5$^{\circ}$).
a) Beam trace of YKA, which was formed by using the theoretical slowness and backazimuth of PP. Arrivals of P, PP, P$^{410}$P and P$^{210}$P are marked by the vertical blue lines.
b) fk-diagram of P time window (u$_P$ = 4.43 s / $^{\circ}$).
c) fk-diagram of signal generated noise in the P coda. No coherent phases are detectable.
d) PP arrival showing the high slowness (u$^{theo}_{PP}$ = 7.65).
e) and f) time windows of P$^{410}$P and P$^{210}$P, respectively. Both fk-diagrams show coherent phases arriving with PP slowness and roughly along the great circle path.

The resolution test displayed show that a small coherent phase which has only 50% to 70% of the amplitude of the incoherent noise can be detected by the sliding-window fk-analysis. Phases arriving simultaneously with different slownesses can be identified and the slowness can be resolved. Phases arriving with travel time differences of 2 s can be separated clearly, independent from the slowness difference of the signals. The sliding-window fk-analysis is able to resolve a backazimuth difference of 10$^{\circ}$ for most noise conditions.


Figure 5.7 shows the sliding-window fk-analysis applied to recordings of an event from North-Halmahera (04-jun-1993 10:49). Figure 5.7a) displays the beam trace of YKA for reference. The beam was formed using the theoretical backazimuth and PP slowness for this event. Theoretical travel times of P, PP, P$^{410}$P and P$^{210}$P are marked as blue vertical lines. The maximum energy of the fk-diagram is marked by the black circle. The theoretical slowness and backazimuth for P (u$_{P}$ = 4.43 s/$^{\circ}$, $\Theta$ = 295.5$^{\circ}$) and PP (u$_{PP}$ = 7.65 s/$^{\circ}$) are marked by the white diamonds. The time window for P can be seen in Figure 5.7b). The P-phase shows a high coherency and u and $\Theta$ for P are well resolved. For reference, a time window from the P coda is added in Figure 5.7c). The fk-diagram contains only low energy and the maximum does not travel along the great circle path. This time window is dominated by uncorrelated phases, most likely due to scattering beneath the array. The coherent PP onset in Figure 5.7d) shows the slowness (u = 7.7 s/$^{\circ}$ which is expected for this phase. For PP the fk-diagram is not as sharp as for P due to the spectral content of P and PP: The spectrum of PP is shifted to longer periods because of the damping of the higher frequencies in the asthenosphere at the surface reflection point. Figures 5.7e) and f) are fk-diagrams of time windows matching travel times for P$^{410}$P and P$^{210}$P, respectively. The travel times are computed using IASP91. Both fk-diagrams show slownesses which identify them as P$^d$P. The backazimuth does not coincide perfectly with the theoretical backazimuth which might be the result of the low amplitude of the precursors. The fk-diagrams show a good fit to the ARF indicating coherent energy. These two time windows fulfil the 3 criteria used to identify P$^d$P. The PP - P$^d$P differential travel times can be converted to depths of the reflectors of 404 km and 210 km using the IASP91 model.
The North-Halmahera event shows a prominent second onset $\sim$18 s after the first onset. The origin of this onset is unknown. The pattern of two separated onsets can be found for PP as well as the precursors, as can be seen in the fk-movie stored on the CD. Consequently, the second onset is most likely a source effect.