Mapping of stress data with geographical coordinates.
Actions affecting stress tensors and using fault data
Input = fault and slip data Output = new stress tensor data
Optimal tensor:
Compute for each fault plane the Euler's angle of the stress tensor best oriented
to reactivate the fault plane, assuming a friction law and using the friction angle φ0.
The geometry can be found in
(Compton, 1966; Etchecopar, 1984; Celerier, 1988 Fig. 6; Tajima and Celerier, 1989; Celerier, 2008 Fig. 1)
Random tensor search:
Monte Carlo approach to search a stress tensor that best explains the slip directions.
This mainly follows the method proposed by Etchecopar et al. (1981) and Etchecopar (1984).
In a first step, stress tensor data are generated by using
a random variable X within [0,1] as follows:
θ = X * 360
φ = Arccos(X)
ψ = X * 180
r0 = X
So that the orientations are uniformly distributed in space.
Note: to obtain a reasonable estimate, a minimum of 1000 random tensors need to be generated.
In a second step, for each tensor, the fault plane error, E(i),
that measures the difference between predicted and observed parameters
for each fault slip data is computed.
In a third step, for each tensor, the global error, F, that measures
its compatibility with respect to the global fault and slip data set,
is computed
In a fourth step the tensors are ranked by increasing value of F and only the n first
tensors are retained (n can be adjusted).
Inversion parameters allow to choose
among a few fault plane error, E(i), and global error, F,
depending on data type.
Inverting data with slip sense only (as in Lisle, 2001) is also an option.
Tensor optimisation.
In this approach one of the 4 parameters (θ, φ, ψ, r0)
defining the stress tensor is varied while the others are kept constant.
The global error, F, is computed as in the random search.
The tensor with the lowest F value is retained.
This can be used to refine solutions found by random search.
Actions using stress tensors and fault data: analysis of the relationship
Input = fault and slip data and stress tensor data
A first step computes for each tensor the fault plane error, E(i), for each fault slip data #i.
A second step computes the global error, F (noted Ftm in the program ouputs), for each tensor and ALL the fault data.
Note: E(i) and F are controlled by the Inversion parameters.
This allows to produce
different graphical representations:
Global plot, all stress tensors and all fault data:
Stereoplots
Histogram of global misfits; Each tensor gives one count in the histogram;
One plot per stress tensor:
Histogram of the angular misfit on each fault plane;
Each fault data gives one count;
Angular misfit as a function of the fault data rank.
Note that the fault data rank are contiguous and correspond to the line number in the data file
(i.e. the order in which the data are read)
NOT NECESSARY the fault data identification numbers that do not need to be contiguous;
Value of s0 to activate each fault plane as a function of the fault data rank;
s0 = (σ1 - σ3)/σ1
is defined as in (Celerier, 1988; Tajima and Celerier, 1989).
Mohr circle where the data are shown with the identification number;
and an output file with tabulated detailed results of stress & fault analysis.
References
Célérier, B., 1988,
How much does slip on a reactivated fault plane constrain the stress tensor ?
, Tectonics, 7, 1257-1278,
doi:10.1029/TC007i006p01257.
Célérier, B., 2008,
Seeking Anderson's faulting in seismicity: a centennial celebration,
Reviews of Geophysics,
46, RG4001, 1-34,
doi:10.1029/2007RG000240.
Compton, 1966,
Analyses of Pliocene-Pleistocene deformation and stresses in northern Santa Lucia range,
California, Geological Society of America Bulletin, 77, 1361-1380.
Etchecopar, A. 1984, Etude des etats de contrainte en tectonique cassante
et simulations de deformations plastiques (approche mathematique). These
d'Etat, Universite des Sciences et Techniques du Languedoc.
Etchecopar, A., Vasseur, G. & Daignieres, M. 1981. An inverse problem
in microtectonics for the determination of stress tensors from fault striation
analysis. J. Struct. Geol. 3, 51-65,
doi: 10.1016/0191-8141(81)90056-09.
Frohlich, C., 1992,
Triangle diagrams: ternary graphs to display similarity and diversity of earthquake focal mechanisms,
Physics of the Earth and Planetetary Interiors, 75, 193-198.
Frohlich, C., 2001,
Display and quantitative assessment of distributions of earthquake focal mechanisms,
Geophysical Journal International, 144, 300-308.
Frohlich, C. & Apperson, K.D., 1992,
Earthquake focal mechanisms, moment tensors, and the consistency of seismic activity near plate boundaries,
Tectonics, 11, 279-296.
Lisle, R., Orife, T. & Arlegui, L., 2001,
A stress inversion method requiring only fault slip sense,
J. Geophys. Res., 106, 2281-2289.
Tajima, F. & Célérier, B., 1989,
Possible focal mechanism change during reactivation of a previously ruptured subduction zone,
Geophysical Journal International, 98, 301-316,
doi:10.1111/j.1365-246X.1989.tb03354.x.