Automatic traveltime picking using the instantaneous traveltime |
syn,ltf0,tt0
Figure 2. (a) Synthetic signal. (b) Local representation of the synthetic signal. (c) using the local map. Contour lines denote values where and the transition through zero is from positive to negative values. The dashed lines in (b) and (c) indicate the effective bandwidth (estimated using equation 11). |
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For most data, the bandwidth of signals is varying with time. It therefore makes sense to estimate the effective bandwidth as a function of time. In order to choose the frequency band automatically, we suggest any data-driven criterion, such as an estimate of the instantaneous frequency or some weighted mean of the frequency. An obvious choice is
The result of the averaging operation is shown in Figure 3(b), in which is averaged over the frequencies in its effective bandwidth. (Figure 3(a) is the same as Figure 2(a) and shown here just to illustrate the agreement between the actual arrivals and the picked arrivals.) Parameter equals the recording (dashed line) at the times of arrival of the spikes, namely at , 1.1991 and 1.3978 s. It should be noted that although there are actually no arrivals between two consecutive arrivals, is not zero in that interval. This is due to the smooth division we employed, when dividing over . Smooth division does not allow the result to vary erratically from sample to sample. Figure 3(c) shows the difference . This difference equals 0 at the times of arrival of the two spikes. However not all zeros of are of interest; rather, only the zero crossings from positive to negative values yield valid traveltimes. Positive maxima of (as in ) denote an imminent arrival of a signal. Let's assume a strong (compared to neighboring events) event at some time . According to equation 3, is a weighted some of the times of arrivals around . The closer is to the imminent arrival at , the more the value of will be biased towards the value . Therefore will be greater than , i.e., . For , the value of will still be biased towards , therefore . In fact, the higher the energy of the arriving signal at some time compared with that of neighboring events, the closer the value of to will be. Accordingly, an event is identified by the non-zero values of the function
syn0,taut0,tau-t0
Figure 3. (a) Synthetic signal (sampling period is 4 ms). Arrivals are at 0.4, 1.2 and 1.4 s. (b) after averaging over the frequencies in the effective bandwidth. The dashed line indicates the recording time. (c) . Zero crossings occur at 0.3996, 1.1991 and 1.3978 s. |
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Automatic traveltime picking using the instantaneous traveltime |