Feasibility of Real-Time Automated Moment Tensor Determination and Path Calibration for 3D Green's Function Construction

Fumiko Tajima, Douglas Dreger, Charles Mégnin and Barbara Romanowicz

Results from feasibility study

Our recent study (Tajima et al., 2001) tested the feasibility of a new automated moment-tensor (AMT) determination system that continuously monitors seismic waveforms from a sparse network of regional broadband stations. This system uses real-time waveforms in a time window that is repeatedly shifted forward with a short time interval ($\sim$20 sec), and unlike the current methods, performs inversion of moment tensors (MT's) over a grid that is set for a region of interest, without prior knowledge of the location and origin time information. The advantage of this system is its independency of the information that is currently provided by dense short-period networks. Significant improvement of the computational efficiency is expected if the system is fully configured on a powerful PC with multiple CPU's of $\sim$1 Gbyte memory, and the inversion over the grid can be updated within the time frame of time window shift ($\sim$20 sec).

Computation speed

In the AMT inversion the time window of waveforms is shifted forward by a prescribed amount $\Delta t$. Therefore, the CPU time needed to compute variance reductions ($VR's$) that is defined by

$${\it VR}\,=\, \left[ 1 \, - {\it Res} \div \left(\, \sum_{k}... ...^k(t)\right) ^2\,dt \right) \right] \times 100 \left(\%\right)$$ (22.1)

where ${\it Res}\,=\, \sum_{k} \int \left(d^k(t) -\, s^k(t) \right)^2 \,dt,$ and $d^k(t)$ and $s^k(t)$ are the data and synthetics at $k$-th station, to evaluate the fitness between data and synthetics for all the grid points and identify the maximum $VR$ to detect an event if any, should be less or equal to $\Delta t$. Roughly speaking, the CPU time is proportional to the number of stations used, and on average, it takes $\sim$1.5 sec to perform inversion for a grid point with varying depths using one station on a 248 MHz Ultra-Sparc 2 workstation. Here, Green's functions for each grid point are read in from the disk, and this I/O process occupies significant portion of the computation time ($\sim$0.4 sec). With this IO it takes $\sim$12 min CPU time to compute $VR's$ over the entire grid (of 160 points) using typically 3 stations. If Green's functions as well as the precomputed cross correlations of Green's functions are stored in the computer memory (Kawakatsu, 1998) and the code is parallelized, the computation time can be cut down substantially. If the AMT is performed on a 10-node cluster of 500 Mhz PC workstations, the computation will be carried out in a much shorter time, somewhere within the time frame of the time shift of $\sim 20$ sec, allowing continuous updates of AMT determination.

Path Calibration

To make the AMT system operational and computationally effective, path calibration is necessary to account for the 3-D structure between virtual sources over the grid and stations. Currently three multi-layered models are used for calculating Green's functions (see the regionalization in Figure 22.1).

Figure 22.1: Regionalization for three multi-layered models, GIL7 (Dreger and Romanowicz, 1994), SoCal (Dreger and Helmberger, 1993), and Mend1 (Tajima et al., 2000).

If Green's functions $G_i^{sk}(t)$ ($s$ stands for a virtual source on the grid, $k$ for a station, and $i$ for the $i$-th elementary moment-tesnsor) are calculated with well constrained velocity models and stored in the memory, the computation time will be shortened significantly, and the efficiency desired for the AMT system can be achieved. However, there is no short cut in calibrating paths for the 3D structure, and this project is still in progress. With Green's functions that account for 3D paths, the gridded structure of the methodology is ideally suited to achieve better resolution and stability.

Computer memory

Another requirement in the AMT is the availability of computer memory. If the horizontal and vertical grid sizes are set to be $0.1^{\circ}$ and 3 km, respectively, there will be

$50 \,\times\,50\,\times\,13\,=\,32,500$ (grid points)

for a $5^{\circ} \times\, 5^{\circ} \times\,39$ km area. The total size of Green's functions for each station requires

$120\, \mbox{(points for 2 min data)}\,\times\, 32,500\ \mbox{(points)}\,$
$\times\,8\,\mbox{(MT elements)}\,\times\,4\,\mbox{(bytes)}\,=\,\sim125\ \mbox{(Mb)}$.

If up to six stations should be used, the memory size will be about 750 Mb, and be slightly increased should the auto correlated Green's functions be also in the memory.

Discussion

Figure 22.2 illustrates the projected time frame of the AMT as compared with the present standard system. The standard system at the BSL as part of the REDI (Gee et al., 1996; 2001) determines MT's for events of $M_{L}\geq 3.5$. The time frame between an event occurrence and MT determination (using 2 different methods) is 8-10 min. If the AMT system monitors seismic wavefield continuously, and updates the regional MT search over the grid every 20 sec, the time interval between the origin time and MT determination could be shortened by several minutes or more.

Effective Use of AMT

A tsunami warning system can be considered for an effective application of the AMT method, as the event location and MT can be determined early enough before the tsunami waves arrive at the coasts. Here, the interval of the P-wave first arrival at the station ($T_{1}$) and the tsunami arrival at the coast ($T_{t}$) is roughly estimated as

$$\Delta T = r_{2}/c -r_{1}/v$$ (22.2)

where $r_{1}$ is the distance between the epicenter and the station, $r_{2}$ the distance between the epicenter and the coast, $c$ is the speed of tsunami wave propagation, $v$ is the P wave speed, and $\Delta T$ is the time interval between the P-wave first arrival and tsunami wave arrival.

Figure 22.2: Timing of event occurrence at $T_{0}$, P-wave arrival at $T_{1}$, grid search for location and MT determination starting at $T_{2}$, and completing at $T_{3}$ in the AMT system, and an event notification arrival by email at $T_{e}$ and MT determination at $T_{mt}$ in the REDI system. $T_{t}$ illustrates tsunami wave arrival time for an epicentral distance of $\sim$200 km.

If $r_{1}$ and $r_{2}$ are roughly $\sim 200$ km, $c$ is $\sim 200$ m/s, and $v$ is $\sim 8$ km/s, then $\Delta T$ is $\sim 975$ sec (or $\sim 16$ min). Since the AMT can characterize the earthquake within several minutes of the P-wave arrival (i.e., during $T_{3}$ - $T_{1}$), there is still $\sim 10$ min after the event location and MT determination, and before the tsunami arrival.

References

Dreger, D. S. and D. V. Helmberger, Determination of source parameters at regional distances with three-component sparse network data, J. Geophys. Res., 98, 8,107-8,125, 1993.

Dreger, D. S. and B. Romanowicz, Source characteristics of events in the San Francisco Bay region, U.S. Geol. Surv., Open-File Rept. 94-176, 301-309, 1994.

Gee, L., D.S. Dreger, D. Neuhauser and B. Romanowicz, The REDI program, Bull. Seism. Soc. Am., 86, 936-945, 1996.

Gee, L., D. Neuhauser, D. Dreger, M. Pasyanos, R. Uhrhammer, and B. Romanowicz, The Rapid Earthquake Data Integration Project, Handbook of Earthquake and Engineering Seismology, IASPEI, in press, 2001.

Kawakatsu, H., On the realtime monitoring of the long-period seismic wavefield, Bull. Earthq. Res. Inst., 73, 267-274, 1998.

Tajima, F., D. Dreger, B. Romanowicz, Modeling of the transitional structure from ocean to continent in the Mendocino region using broadband waveform data, EOS Trans. Am. Geophys. Union, 81, 2000.

Tajima, F., C. Mégnin, D. Dreger, and B. Romanowicz, Feasibility of real-time broadband waveform inversion for simultaneous moment tensor and centroid location determination, Bull. Seism. Soc. Am., in press, 2001.

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