Paleostress inversion refers to the determination of
paleostress history from evidence found in rocks, based on the principle that past
tectonic stress should have left traces in the rocks. Such relationships have been discovered from field studies for years: qualitative and quantitative analyses of deformation structures are useful for understanding the distribution and transformation of
paleostress fields controlled by sequential tectonic events. Deformation ranges from microscopic to regional scale, and from
brittle
A material is brittle if, when subjected to stress, it fractures with little elastic deformation and without significant plastic deformation. Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Br ...
to
ductile
Ductility is a mechanical property commonly described as a material's amenability to drawing (e.g. into wire). In materials science, ductility is defined by the degree to which a material can sustain plastic deformation under tensile stres ...
behaviour, depending on the
rheology
Rheology (; ) is the study of the flow of matter, primarily in a fluid ( liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an ap ...
of the rock, orientation and magnitude of the stress etc. Therefore, detailed observations in outcrops, as well as in
thin section
In optical mineralogy and petrography, a thin section (or petrographic thin section) is a thin slice of a rock or mineral sample, prepared in a laboratory, for use with a polarizing petrographic microscope, electron microscope and electron ...
s, are important in reconstructing the
paleostress trajectories.
Inversions require assumptions in order to simplify the complex geological processes. The
stress field is assumed to be spatially uniform for a faulted
rock mass and temporally stable over the concerned period of time when faulting occurred in that region. In other words, the effect of local fault slip is ignored in the variation in small-scale stress field. Moreover, the maximum shear stress resolved on the
fault surface from the known stress field and the slip on each of the fault surface has the same direction and magnitude. Since the first introduction of the methods by Wallace and Bott
[Bott, M. H. P. 1959. The mechanisms of oblique slip faulting. Geol. Mag. 96,109-117.] in the 1950s, similar assumptions have been used throughout the decades.
Fault slip analysis
Conjugate fault system
Anderson was the first to utilize conjugate fault systems in interpreting paleostress, including all kinds of conjugate faults (normal, reverse and strike-slip). Regional conjugate fault can be better understood by comparison to a familiar rock mechanics experiment, i.e. the Uniaxial Compressive Strength (UCS) Test. Basics of their mechanisms are similar except the principal stress orientation applied is rotated from perpendicular to parallel to the ground. The conjugate fault model is a simple way to obtain approximate orientations of stress axes, due to the abundance of such structure in the upper brittle crust. Therefore, a number of studies have been carried out by other researchers in assorted structural settings and by correlating with other deformation structures.
Nonetheless, further development revealed the deficiency of the model:
::1. Important geometrical properties absent in practical situation
The geometrical properties of conjugate faults are indicative of the sense of stress, but they may not appear in the actual fault patterns.
*
Slickenside lineations normal to fault plane intersection
* Symmetrical sense of motion that gives the obtuse angle in the direction of lengthening
* Relation between the intersecting angle of fault planes and mechanical properties, with reference to information from
rock mechanics Rock mechanics is a theoretical and applied science of the mechanical behavior of rock and rock masses; compared to geology, it is that branch of mechanics concerned with the response of rock and rock masses to the force fields of their physical env ...
experiments in lab
::2. Observed fault patterns are far more sophisticated
There are often oblique pre-existing faults, planes of weaknesses or striations to the fault slip, which do not belong to the conjugate fault sets. Neglecting this considerable amount of data would cause error in analysis.
::3. Neglecting the stress ratio (Φ)
This ratio provides the relative magnitude of the intermediate stress (σ
2) and thus determines the shape of the stress ellipsoid. However, this model does not give an account on the ratio, save for some specific cases.
Reduced stress tensor
This method was established by Bott
in 1959, based on the assumption that direction and sense of slip occurs on the fault plane are the same with those of the maximum resolved shear stress, hence, with known orientations and senses of movements on abundant faults, a particular solution T (the reduce stress tensor) is attained.
It gives more comprehensive and accurate results in reconstructing paleostress axes and determining the stress ratio (Φ) than the conjugate fault system. The tensor works by solving for four independent unknowns (three principal axes and Φ) through mathematical computation of observations of faults (i.e. attitude of faults and lineations on fault planes, direction and sense of slip, and other tension fractures).
This method follows four rigorous steps:
# Data Analysis
# Computation of Reduced Stress Tensor
# Minimization
# Check of Results
Data analysis
Reconstruction of paleostress requires large amount of data to attain accuracy, so it is essential to organize the data in comprehensible format prior to any analysis.
:1) Fault Population Geometry
Attitude of fault planes and slickensides is plotted on rose diagrams, such that the geometry is visible. This is particularly useful when the sample size is enormous, it provides the full picture of the region of interest.
:2) Fault Motion
Fault movement is resolved into three components (as in 3D), which are vertical transverse, horizontal transverse and lateral components, by trigonometric relation with the measured dips and trends. Net slip is shown more clearly which paves the way to understanding the deformation.
:3) Individual Fault Geometry
Fault planes are represented by lines in
stereonets (equal area lower hemisphere projection), while
rakes on them are indicated by dots sitting on the lines. It helps to visualize the geometrical distribution and possible symmetry among individual faults.
:4) P
(pressure) and T
(tension) Dihedra
This is a concluding step of compiling all the data and check their mechanical compatibility, also could be seen a preliminary step in determining major paleostress orientations. As this is a simple graphical representation of the fault geometry (being the boundaries of dihedra) and sense of slip (shortening direction indicated by black and extension depicted by grey), while it is able to provide good constraints on the orientation of principal stress axes.
The approximation is built upon the assumption that the orientation of maximum principal stress (σ
1) most probably passes through the greatest number of P-quadrants. Since fault plane and auxiliary plane perpendicular to striations are considered the same in this method, the model can be directly applied to
focal mechanism
The focal mechanism of an earthquake describes the deformation in the source region that generates the seismic waves. In the case of a fault-related event it refers to the orientation of the fault plane that slipped and the slip vector and ...
s of earthquakes. Nonetheless, due to the same reason, this method cannot provide accurate determination of paleostress, as well as the stress ratio.
Determination of paleostress
= Reduced stress tensor
=
Stress
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
can be considered as a matrix with nine components being the nine stress vectors acting on a point, in which the three vectors along the diagonal (highlighted in brown) represent the principal axes.
The reduced stress tensor is a mathematical computation approach to determining the three principal axes and the stress ratio, totally four independent unknowns, calculated as eigenvectors and eigenvalue respectively, so that this method is more complete and accurate than the mentioned graphical approaches.
There are a number formulations that can reach the same final results but with distinctive features:
(1)
,
where
, such that
. This tensor is defined by setting σ
1, σ
2 and σ
3 as 1, Φ and 0 (highlighted in pink) respectively, due to choosing
and
as the mode of reduction. The advantage of this formulation is the direct correspondence to stress orientation, thus the stress ellipsoid, and the stress ratio.
(2)
This formulation is a deviator, which requires more computation to obtain information of the stress ellipsoid despite maintaining a symmetry in mathematical context.
[Angelier, J. 1984. Tectonic Analysis of Fault Slip Data Sets. Journal of Geophysical Research, 89, B7, 5835-5848.]
= Minimization
=
Minimization aims to reduce the differences between the computed and observed slip directions of fault planes by choosing a function to proceed the least square minimization. Here are a few examples of the functions:
(1)
The very first function used in fault slip analysis does not account on the sense of individual slip, which means altering the sense of a single slip does not affect the result. However, individual sense of motion is an effective reflection of orientation of stress axes in real situation. Hence, S
1 is the simplest function but include the importance of sense of individual slip.
(2)
S
2 is derived from S
1 based on variation in computational process.
(3)