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Proceedings of the National Academy of Sciences of the United States of America
FIG. 1. Example of geometric incompatibility near fault junction. 1, Faults; 2, horizontal component of absolute movement of a block, ; 3, horizontal component of slip rate on a fault, ; A, B, C, and D, corners of the blocks; a, initial position of the blocks; b, extrapolation of initial movement. Corners A and C are converging and would overlap; this indicates that the movement cannot be realized without the change of the fault geometry.
tion and deformation of blocks separated by the faults. Both measures of incompatibility have the following useful property, reminiscent of the Stokes formula: To estimate their values in a whole region, it is sufficient to know only the movements on the faults that cross the boundary of this region.
Analytical expressions for G are derived below first for a single junction and then for a system of faults with many junctions within a given contour. The expressions for a single junction are the same for rotating and nonrotating blocks. For a fault system with many junctions these expressions are
FIG. 2. Example of geometric incompatibility. Notations are the same as in Fig. 1. Geometric incompatibility leads to divergence of the block corners.
different for deformable/rotating blocks; we consider this case in the Appendix.
Geometric Incompatibility for a Single Junction
Consider n fault segments with a common junction point; a fault crossing this point is regarded as two segments. We choose some segment as the first one, and assign the numbers from 2 to n to the other segments counterclockwise. The n blocks separated by these faults meet at the junction. We number them in such a way that the ith fault separates the blocks i and i+1.
Let be the rate of movement of the ith block at the junction point, and the rate of relative movement on the ith fault. We always set and Because we consider only horizontal components of the movements, the vectors and are two-dimensional. For two such vectors, and , we define the cross-product . After a time period t the junction, which we place at the origin, becomes, due to block movement, a polygon with the vertices . Computing the (oriented) area of this polygon, we obtain the following expression for the geometric incompatibility G defined in ref. 2:
We easily verify that this sum does not change when we add a common vector to all the rate vectors . Replacing by and substituting , we can rewrite G in terms of the relative movement rates :
This formula depends on the arbitrary choice of the first segment. However, it will lead to the same values of G, for any choice, if if the observed values of satisfy the Saint Venant condition Eq. 1. If this condition is not satisfied— e.g., due to observational errors or block deformation, we have to satisfy it by modification of . This can be done in a nonunique way. In the absence of additional information we simply subtract from each of a fraction of proportional to the absolute value of vi:
The following properties of G may be of interest: (i) The value of G does not change when all slip rates reverse directions, (ii) For intersections of strike-slip faults, a negative sign of G indicates the tendency of the blocks to “penetrate” each other at the junction (Fig. 1), so that the movement along one fault locks up another fault. On the contrary, if G were positive, the movement on one fault unlocks another one (Fig. 2).
Geometric Incompatibility for a System of Faults with Multiple Junctions
We consider here the case of rigid nonrotating blocks, which is analytically simpler than the general case and can be of independent interest. Generalization for the case of deformable and/or rotating blocks is given in the Appendix.
Suppose we have a map with a system of blocks divided by faults. Consider on this map a simply connected region D surrounded by a boundary L. Because the blocks do not rotate, the rates of block movement are the same at all points of a block, and the rates of relative fault movements are the same between fault junctions.
As before, we number the blocks crossing the boundary L of the region D counterclockwise, and number the faults crossing