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Geological diagrams for 3D motion

Science, May 16, 2003, shows some interesting geological diagrams for describing 3D slippage. Can someone who knows the field please provide a careful explanation of the ball diagrams in these figures?

Figure 2 from David Bowman, Geoffrey King, Paul Tapponnier, "Slip Partitioning by Elastoplastic Propagation of Oblique Slip at Depth", Science, Vol. 300 (May 16, 2003), pp. 1121-1123.

Figure 2 from Donna Eberhart-Phillips, et al., "The 2002 Denali Fault Earthquake, Alaska: A Large Magnitude, Slip-Partitioned Event", Science, Vol. 300 (May 16, 2003), pp. 1113-1118.

-- Edward Tufte

For lack of the "careful explanation" requested above, here is a page that outlines the analysis behind these lower hemisphere projection diagrams along with my layman's best guess at what's going on:

Seismic activity results from the actions of two adjacent and rigid regions along a plane that divides them. This is the "real" fault plane, and it runs through the focus of the event. An "auxiliary" plane also runs through the focus- it's indistinguishable, but is used to divide the region into quadrants.

On opposite sides of the real fault plane, the ground particles are moving in different directions and are in two different regions, while the auxiliary fault plane separates compression from dilation within a single region. With seismograph readings, it's possible to determine the orientation of these planes.

The "beach ball" diagrams show how the 2 planes intersect a sphere, while the "half-melons" in Figure E show how the planes are projected onto a lower hemisphere, resulting in the stick diagrams on Maps B and D.

Can any seismologists help out here?

-- Matt Frost (email)

I am not familiar with the diagrams in Figure 1, but the ball diagrams in Figure 2 are standard presentations of earthquake focal mechanisms. Earthquakes are caused by motion along faults. For a quick analog, join your hands flat as if praying and make your index fingers horizontal. Now slip one hand relative to the other in a horizontal motion; you just generated an earthquake along a vertical fault (the plane between your hands). This type of fault is called a strike-slip fault; an example is the San Andreas fault.

Seismic waves can be measured at great distances from the source in a network of stations around the globe. Join your hands again and imagine that there are seismic stations in the horizontal plane that measure the first motion of the earthquake, meaning whether the first wave arriving from the earthquake made the ground move away from the earthquake source or toward it. Suppose you moved your right hand forward relative to the left. Then hypothetical seismic stations would make the following observations of first motion:

  • A station in front of you and to your right: first motion away from the source;
  • A station behind your back and to your right: first motion toward the source;
  • A station behind your back and to your left: first motion away from the source;
  • A station in front of you and to your left: first motion toward the source.
The beach balls in Figure 2 are a simple graphical way to present these first motion observations. They are obtained by projecting on a sphere first motion observations from a global network of seismic stations and coloring in black the quadrants where the first motion was away from the source and in white the quadrants where the first motion was toward the source. The three balls in Figure 2 that look like a BMW logo show strike-slip motion as in the joined hands example, with the ground on the north side of the Denali fault moving to the right (east) relative to the south side. The ball with a central black sector and two white sectors on the sides instead shows a reverse fault mechanism, where the fault plane is inclined and the fault is the result of convergence between the blocks on the two sides of the fault. A normal fault, where the two blocks move away from each other, would show the negative pattern (a central white sector and two black sectors).

This is hard to explain without pictures. For a detailed treatment with plenty of diagrams, see chapter 6 of Cox, A. and Hart, R.B., "Plate Tectonics: How it Works," Blackwell Scientific Publications, Palo Alto, California, 1986.

-- Alberto Malinverno (email)

A good illustration of the focal mechanism "beach balls" is at

-- Alberto Malinverno (email)

The two types of beachball diagrams shown represent two distinctly different, but related, kinds of information. The lower diagram is a fairly standard earthquake focal mechanism diagram, while the upper diagram represents a novel attempt to show details of 3D slip patterns on sets of conjugate faults. I will only briefly touch on the focal mechanism diagrams, as they are described well in many other places (including the first two answers), and then I will comment more on the strain diagrams, which are both novel and content-rich.

The earthquake focal mechanism diagrams in the latter figure are quite standard representations of the distribution of first arrivals recorded at seismographs. Typically, the black quadrants represent compressional first arrivals (ground first moves up), while the white quadrant represents dilational first arrivals (ground first moves down). If you think about the relative directions of motion of two patches of rock across a planar fault, it is easy to see how the waves sent out from the event will partition into quadrants. For each side there will be 90 degrees in front over which the material will appear to be appraoching an observer and 90 degrees in back where the material will be pulling away. In the case of a large number of seismographs recording an earthquake it is usually fairly easy to establish the orientation of the compressional and dilational quadrants and the only real challenge is figuring out which of the two dividing planes they define represents the actual fault along which the slippage occurred.

The information conveyed by these diagrams can be very valuable and they serve to graphically represent the orientation of the principle stresses which are driving the deformation, with the principle shortening direction located in the middle of the white quadrant. A geoscientist presented with the focal mechanisms in a particular area can quickly assess the orientation of the three principle stresses active in the region and the types of faulting they represent. If the white quadrant is facing up, then the deformation is dominated by horizontal extension and normal faulting. Similarly, a black quadrant facing up indicates horizontal compression and thrust faulting. If the focal mechanisms are showing an X pattern, then they indicate horizontal shear or strike-slip faulting. These types of displays have been routine in regional studies of crustal deformation patterns for decades.

The beachball diagrams in the upper figure are an attempt to illustrate a complex deformation pattern in a simplified way. In the deformation of a body of rock there are typically a set of conjugate shears which act to make up the total deformation. Andersonian faulting theory suggests these conjugate shears should be oriented in predictable ways with respect to the principle stresses driving the deformation. The two orientations will be planes that include the intermediate stress direction and sit 30 degrees on either side of the principle stress direction (imagine the surfaces of a wedge used to split wood). The actual slip directions on the conjugate faults will be defined by the intersection of the shear planes with the plane defined by the minimum and maximum stresses.

The paper from which the illustration was taken is trying to illustrate the variation in deformation orientation in the vicinity of very large faults (tectonic plate boundaries). To do this they needed a way to show how the orientation of the conjugate shears varies as you approach the fault. They settled on a symbology for their maps where each of the conjugate shears is indicated by a line and a dot. Each line represents the orientation of the intersection of the shear plane with a horizontal surface and the dots represent the orientation of the slip direction on the shear plane. The beachball diagrams simply illustrate how they derive the appropriate positions for the dots with respect to the lines. The dots represent the intersection of the slip line with a lower hemisphere plot, projected vertically onto the horizontal plane. The authors attempt to convey additional information by the use of color, with red and blue indicating opposite senses of shear and light and dark shades indicating the steepness of the slip line.

While there is a lot of information conveyed in the symbols the authors have chosen, there is a lot of duplication of information and the graphical format does not easily convey the information content to the viewer. The same information could have been conveyed by a variant of the focal mechanism beachballs decrscibed earlier. If you create a similar beachball with unequal "quadrants" of 60 and 120 degrees, then you could convey all the same information in a much more approachable and intuitive way. The divisions of normal, thrust and strike-slip environments (and hybrid cases) would be just as immediately evident as in the focal mechanism diagrams. Because the slip orientations are fixed with respect to the geometry of the conjugate shears, this information would still be communicated. There would be no need for color and shades to convey sense of shear or steepness of dip, as this would all be implicit in the beach ball geometry.

The resulting simplicity would actually help convey the intended information. The current cloud of multicolored lines and dots overwhelms the visual senses and obscures the underlying pattern. The difficulty in comprehending the information conveyed - which spurred the question which kicked off this thread - suggests I am not the only one who found this graphical formula difficult to absorb. I feel this is a case where simplicity of form might have more effectively communicated the pattern observed.

-- Scot Krueger (email)

That is a very helpful and informative analysis!

-- Edward Tufte

Not quite on topic, but it's the closest thread I could find. The New York Times recently (April 11, 2006) published a very good map of the Great San Francisco Earthquake:

The image combines a 3-dimensional map of quake intensity (laitude, longitude, intensity) with 2-d fault lines, urban areas, and slip along the San Andreas Fault. The pastel colors facilitate the layeringof information. Hill shading would be a nice addition, but possibly difficult to include at web resolution. Hopefully the links won't disappear behind the 'TimesSelect' firewall.

As a side comment: would it be possible to add a thread for users to post examples of superlative imagery?

-- Rob Simmon (email)

Threads relevant to maps:

Threads relevant to nature studies:
Theoretical speculations on leaving the flatland of paper and computer screens and working now in real-land and space-land.

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