Joor Loohuis' Pages

Stress Inversion

Simultaneous focal mechanism and stress tensor inversion using a genetic algorithm

J. Loohuis and T. van Eck

A more comprehensive version of this article has been published in Physics and Chemistry of the Earth, Vol. 21, no. 4, pp. 267 - 271, 1996.

outline | publications | presentations | references

Abstract:A new inversion program is presented that determines the regional stress tensor and a set of focal mechanisms simultaneously, using P-wave first motions as input data. The inversion procedure finds a reduced stress tensor, expressed in three Euler angles and a shape factor, and a set of fault planes. The following constraints are applied:

all examined earthquakes are caused by the same stress tensor, and
the slip on the fault planes is parallel to the direction of maximum shear stress

For this highly nonlinear problem a nested genetic algorithm was implemented. The measure of misfit was determined using a weight for the observation quality and objective estimates of the errors in the azimuths and the take-off angles of the arrivals. The program performance was illustrated in two examples. One was based on a synthetic data set, which allowed testing of the algorithm and determination of the optimal inversion parameters. The other was based on a data set which comprised a selection of events from an earthquake cluster near the Jordan-Dead Sea transform fault.

outline

In theory the method is rather simple. Given a certain reduced reduced stress tensor (where the hydrostatic part has been taken out of the full stress tensor, and the remainder has been scaled), the direction of maximum shear stress on a fault plane is determined. The assumption is made that slip during an earthquake will take place in this direction. This is made clear in fig. 1. The green vector represents the normal on the fault plane.	Due to the regional stress field, expressed in a tensor, a stress vector (red) acts on the plane. This stress vector can be decomposed in a part normal to the fault plane and a shear stress (orange) situated in the fault plane. If both the slip direction and the orientation of the fault plane are available, the radiation pattern for a double couple source mechanism can be synthesized.This formulation is the forward problem.

	Fig. 1.The relation between stress directions and slip on a fault plane (Bott's Hypothesis) click on the image to obtain a larger one...

The inverse problem can be defined as the determination of a reduced stress tensor and a fault normal which demonstrate the best fit to an observed radiation pattern. If it is further assumed that the regional stress field causes all seismic events in a restricted region and a restricted time interval, the inverse definition is extended to the situation where one reduced stress tensor causes slip on different fault planes, resulting in a variation of source mechanisms.	The object of the method we designed is to invert for one regional stress tensor and a set of fault normals directly from raw measurements in the form of P wave first arrival polarities. Since the problem is strongly nonlinear, we resorted to a simple genetic algorithm for the optimization. This method has been applied to a set of 27events that took place close to the Arava transform fault south of the Dead Sea in Israel. Some results are plotted in the figures below.

Fig. 2. Solution for the maximum compression direction (a) and the sampling of the same (b) , for the Arava data set. click on the image to obtain a larger one...

Above the confidence levels of 68% and 95% for the compression axis are plotted in a stereographic frame. The black dot at 338/00 indicates the best solution.	It is obvious that there are several optima, of which at least two are significant. On the right the sampling of the same axis is plotted. Due to the nature of the genetic algorithm, most orientations have been sampled more than once.

	Fig. 3. Fault plane solutions for the 27 events of the Arava data set for the best overall solution. click on the image to obtain a larger one...

As mentioned, a set of fault orientations is obtained together with the reduced stress tensor. The plots above show the fault planes for the best solution of the Arava cluster. In cyan the fault planes are plotted (lower hemisphere),	and in red the slip vectors as predicted by the stress model are indicated. Closed symbols stand for normal faulting, open symbols (2) denote thrust faulting. The source mechanisms have been grouped by visual inspection.

publications

Camelbeeck, T., T. van Eck, R. Pelzing, L. Ahorner, J. Loohuis, H.W. Haak, P. Hoang-Trong and D. Hollnack, The earthquake of April 13, 1992 near Roermond, the Netherlands, and its aftershocks, Geologie en Mijnbouw, 73, 181-197, 1995.
Loohuis, J. and T. van Eck, Simultaneous focal mechanism and stress tensor inversion using a genetic algorithm, Physics and Chemistry of the Earth, 21, no. 4, 267 - 271, 1996.

presentations

Loohuis, J. and T. van Eck, Joint inversion of stress tensor and focal mechanisms for Roermond aftershocks, Journees Luxembourgeoises de Geodynamique, 77th session, November 1994, Walferdange, Luxemburg (abstract) .
van Eck, T., J. Bolte, J. Loohuis and S. Yoshioka, The Roer Valley earthquake of April 13, 1992: An intraplate earthquake with distant aftershocks, IUGG XXI General Assembly, July 1995, Boulder, Colorado, U.S.A.
Loohuis, J. and T. van Eck, Simultaneous focal mechanism and stress tensor inversion using a genetic algorithm, EGS, May 1996, The Hague, The Netherlands. ( abstract).

references

Angelier, J., Tectonic analysis of fault slip data sets, J. Geophys. Res., 89, 5835, 1984.
Arfken, G., Mathematical methods for physicists, 3rd edition, pp. 198-200, Academic Press, Inc, San Diego, Cal., 1985.
Brillinger, D. R., A. Udias and B. A. Bolt, A probabilistic model for regional focal mechanism solutions, Bull. Seismol. Soc. Am., 70, 149, 1980.
Dillinger, W. H., S. T. Harding and A. J. Pope, Determining maximum likelihood bodywave focal plane solutions, Geophys. J. Royal Astr. Soc., 30, 315, 1972.
Gephart, J. W., and D. W. Forsyth, An improved method for determining the regional stress tensor using earthquake focal mechanism data: application to the San Fernando earthquake sequence, J. Geophys. Res., 89, 9305, 1984.
Gephart, J. W., FMSI: a FORTRAN program for inverting fault/slickenside and earthquake focal mechanism data to obtain the regional stress tensor, Computers and Geosciences, 16, 953, 1990.
Goldberg, D. E., Genetic algorithms in search, optimization and machine learning, Addison-Wesley, 1989.
Kobayashi, R., and I. Nakanishi, Application of genetic algorithms to focal mechanism determination, Geophys. Res. Lett., 21, 729, 1994.
Parker, R. L., and M. K. McNutt, Statistics for the one-norm misfit measure, J. Geophys. Res., 85, 4429, 1980.
Pope, A. J., Fiducial regions for body wave focal plane solutions, Geophys. J. Royal Astr. Soc., 30, 331, 1972.
Ranalli, G., Rheology of the Earth, 2nd edition, pp 20-23, Chapman & Hall, London, UK, 1995.
Rivera, L. A., and A. Cisternas, Stress tensor and fault plane solutions for a population of earthquakes, Bull. Seismol. Soc. Am., 80, 600, 1990.
Sambridge, M., and G. Drijkoningen, Genetic algorithms in seismic waveform inversion, Geophys. J. Int., 109, 323, 1992.
Sambridge, M., and K. Gallagher, Earthquake hypocenter locations using genetic algorithms, Bull. Seismol. Soc. Am., 83, 1467, 1993.

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