Vespagrams

The comparison of the P$^d$P polarity with the PP polarity is used to determine the sign of the impedance contrast at the upper mantle discontinuities. Due to the knowledge of the impedance contrast at the surface, the impedance change at the discontinuity can be determined. P$^d$P is too small to be identified in the unprocessed seismograms. However, the polarity of the wavelets cannot be analysed with the fk-analysis. Therefore, another method to study P$^d$P is necessary which preserve the polarity information.
The vespa method ( Velocity Spectral Analysis) is used to increase the P$^d$P signal amplitudes. The vespa-method analyses the seismic signals recorded at an array in terms of the energy content of the incoming signal as a function of the slowness (Davies et al., 1971). The seismic waves are assumed to travel along one specific backazimuth. Mostly, the backazimuth of the theoretical great-circle path from source to receiver is used. The power is mapped as a function of slowness and the array is steered along the backazimuth of the great-circle path. The seismograms of the array are stacked. The output of the array, $v_u(t)$, depending on the slowness, $u$, is calculated by:

$$v_u(t) = \frac{1}{M}\sum\limits_{i=1}^{M} s_i(t-t'_{u,i}),$$ (13)

with    $s_{i}$(t)    :of station $i$

		$t'_{u,i}$		:		 relative travel time to station $i$ for slowness $u$

		$M$		:		 number of array stations.

The summed traces of the array are displayed as a function of slowness. The resulting diagram is called a vespagram.
For this study a variation of the vespa method, the nth-root vespa (Muirhead and Datt, 1976), is used. The nth-root vespa provides a higher slowness resolution than the traditional vespa-method.
The nth-root vespa takes the nth-root (n=2,3,4,...) of the amplitudes of all array traces before the summation. The absolute value of a single trace is taken and the sign of each sample is saved.

$$v'_{u,n}(t) = \frac{1}{M} \sum\limits_{i=1}^{M} \left\vert s... ...})\right\vert^{\frac{1}{n}} \cdot signum\left\{s_i(t)\right\}.$$ (14)

After the summation, the vespagram traces are raised to the nth power and the saved signs of the samples are multiplied:

$$v_{u,n} = \left\vert v'_{u,n}(t)\right\vert^{n} \cdot signum\left\{v'_{u,n}(t)\right\}.$$ (15)

The nth-root vespa gives greater importance to the coherency of the signal than the linear vespagram taking into account the similarity of amplitudes. Therefore, incoherent noise is reduced effectively. On the other hand, the nonlinear processing (nth-root) deforms the wave forms of the seismic waves from continuous signals to spikes. However, due to the conservation of the signs of the samples, the nth-root vespagram can be used to study the impedance contrast at the discontinuities using P$^d$P.
For this purpose, a P$^d$P phase found by the fk-analysis is isolated in the nth-root-vespagram and compared with the PP onset in the vespagram.
Figure 5.8 shows as example the North Halmahera event (04-jun-1993 10:49) recorded at YKA. The filtered seismograms of the YKA stations are shown in Figure 5.8a). A 400 s time window is displayed, starting $\sim$40 s before the P arrival and ending $\sim$100 s after the PP arrival. The arrival times of P and PP are marked. A very strong second onset $\sim$18 s after the first P onset is visible. This phase shows higher amplitudes than P and is most probably a result of the source mechanism. A weak onset $\sim$40 s before the PP can be identified.
The 4th-root vespagram for the same time window is displayed in Figure 5.8b). The vespagram was computed using the backazimuth of the great circle path between source and receiver. The vespagram displays the slowness along the y-axis over time. The travel times of P and PP are marked on the x-axis and the appropriate slowness for these two phases are labelled (u$_{PP}$= 7.75 s/$^{\circ}$ and u$_P$ = 4.43 s/$^{\circ}$). Two phases with slownesses between 7.4 s/$^{\circ}$ and 7.8 s/$^{\circ}$ are visible in the vespagram. In addition to the phase $\sim$40 s before the PP arrival already visible in the seismogram, a phase $\sim$85 s before PP can be identified. Identification of a precursor in the vespagram of a small aperture array is questionable, because the vespagram is only computed for the theoretical backazimuth. A phase arriving at the array with a different backazimuth than the one used for computation of the vespagram can produce a wrong slowness for this phase (Jahnke, 1998). Because of this uncertainty of vespagrams for the study of the P$^d$P phases the sliding-window fk-analysis is used.

Figure 5.8: a) Seismogram examples of the event under North Halmahera (04-jun-1993 10:49). Source parameters as described in Figure 5.7. A 400 s time window of the velocity vespagrams starting 40 s before the P arrival is displayed. The amplitudes are normalized to the PP onsets. The travel times for P and PP are marked. A second arrival $\sim$18 s after the P arrival is most likely a result of the source mechanism. A weak onset before the PP arrival is visible.
b) 4th-root vespagram of the same event. The same time window is displayed. The slowness steps are 0.2 s/$^{\circ}$. The theoretical backazimuth $\Theta$ = 295.5$^{\circ}$ was used for the computation of the vespagram. The slownesses and travel times of P and PP are marked. Two phases with the corresponding slownesses and travel times of P$^d$P can be seen in the time window before PP. These phases can be identified as P$^d$P phases from the 410 and the L by the sliding-window fk-analysis.