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Fig 1. Coyote Buttes, Utah, USA

Perceiving the true dimensions of crossbed strata can be tricky; what appears upon an erosion surface (*apparent* dimensions) is often quite different from true dimensions.

Sedimentary strata often end up as inclined tabular planes, either through tectonic forces or, as in case of aeolian cross-strata, having been deposited that way. **Dip**, or dip angle, is the angle of the feature's incline from horizontal; so a horizontal surface would have a dip of 0°, a feature tilted downward 30° has a dip of 30°, and a vertical feature dips 90°. The **dip direction** is the compass direction (also called **azimuth**) toward which the feature dips; for a plane, it's the direction water would run if poured upon the inclined plane.

An inclined plane makes a line where it intersects with another plane. The line of intersection of an inclined plane and a horizontal plane is its **strike line**, or line of strike. The **strike** of a line is the compass angle (azimuth) of its bearing (ie., direction). From these definitions, it follows that the strike of an inclined plane is always perpendicular to that plane's direction of dip.

Features that look like lines are **lineations**. Cross-strata often has many parallel pinstripe planes that appear as parallel lines on a planar erosion surface. The orientation of an inclined line is described by its dip and dip direction; when referring to a line, the terms **plunge** and **trend** are often used instead of dip and dip direction.

Fig 2. Apparent strata dip and thickness vary with exposure plane angle

The lines on a planar erosion surface are the result of that erosion surface intersecting with planar strata within an outcrop. As will be discussed below, the dip of those erosion surface lines is usually not the same as the **true dip** of the strata within the outcrop; the dip of the surface lines is the misleading **apparent dip** of the strata. Said differently: The true dip of a *line* on an erosion surface is not necessarily the true dip of the *strata associated with that line*. Similarly, the **true thickness**, as measured perpendicular to its bounding surfaces, is usually not the same as the distance on the erosion surface between the lineations that bound that strata (the **apparent thickness**) as it appears on an obliquely intersecting erosion surface.

Wikipedia links: Strike and dip, azimuth, lineation; other: Aeolian dune terminology

It's easy to forget that what is visible upon the erosion surface is not always an orthogonal 'true dimensions' view of the strata, especially when, as for cross-strata, the strata are not horizontal.

To get a feel for how erosion surfaces and cross-strata interact, use the two sliders in the graphic below to explore what patterns can arise on a planar erosion surface that cuts a block of idealized crossbedded sandstone.

The block has four horizontal sets of crossbedding (two full and two partially truncated). All strata dip in the same direction, and are tabular planar with a bottom wedge of wind ripple. The sliders control the dip (left slider) and dip direction (bottom slider) of an 'erosion surface' that cuts through the crossbedding. The crossbed can be viewed from any direction by clicking on the block and dragging.

Many orientations of the erosion surface give rise to patterns on the erosion surface (the only surface visible, of course, at an outcrop) that look like quite different from those in profile view (ie., the view parallel to slipface strike).

Crossbed strata appear on an erosion surface with true thickness when the erosion surface is perpendicular to the tabular planar strata, and with true dip when the erosion surface is perpendicular to the crossbed strike direction.

Fig 3. Buckskin Gulch

Both of these conditions are met when the erosion surface is vertical (left slider set to 90° dip) and perpendicular to strike (lower slider set to 90°) (also click 'Erosion dip' to be sure the erosion surface is being displayed).

Vertical exposure alone doesn't guarantee true dip and thickness; the surface must also be perpendicular to strike. At all other angles, strata appear thicker than they really are, and strata dips appear shallower than they really are.

Simulate walking through a slot canyon with vertical walls (eg., fig 3) by clicking 'Lock', setting the erosion surface dip to 90°, and moving the lower slider. The change of dip is quite pronounced away from perpendicular; the change of thickness is more subtle (because the slipface dip is low; you can see this angle by rotating the block perpendicular to strike). Thus 'close to perpendicular' is often good enough to get a sense for true dimensions and dip.

Fig 4. Shaded area is

perpendicular to view and crossbed; bright area is pattern on an oblique erosion surface

If, as is often approximately the case, the crossbed strata are laterally (along their strike) identical, as in the idealized case of tabular planar strata, a cross-sectional view of the crossbeds in true dimensions can be obtained by looking at the erosion surface from an angle that is parallel to the strike of the crossbeds. Press 'Slipface strike', and set the 'Lens' slider to 250mm, to see such a view; notice that you can then move the erosion slope (using the other sliders) and yet the strike-parallel view of the erosion slope pattern is pretty much a continuation of the vertical perpendicular section of the side of the cube (fig 4).

An oblique view such as that is made up of a sequence of infinitessimally-thin parallel cross-sections. As long as each cross-section is nearly identical, the aggregated sequence has the appearance of a single dip-parallel cross-section, in true dimensions. To get a feel for how neighbouring cross-sections contribute to the view, click the 'Show contributions' checkbox and then explore with the sliders.

Crossbedding is often wonderfully uniform at meter-scales, so a strike-parallel view often is quite informative about the shape of crossbeds. However, in situations were the crossbedding is not laterally uniform, e.g., curved or deformed, of course the synthetic view will be different from any single dip-parallel cross-section. It's usually obvious to what extent one can rely upon a strike-parallel view.

There's a more common barrier to obtaining a strike-parallel view: perspective distortion, discussed next.

To our eyes and cameras, parallel lines with any depth component (eg., traveling toward the horizon, or on a sloping surface) appear to be lines intersecting at a point (a 'vanishing point'); that's *perspective view*. That doesn't happen in *orthogonal view*, where railway tracks and roads stretching off to the horizon would maintain their width all the way. Perspective view preserves lines as lines, but (unless perpendicular to view) changes their slope. Our brains seem to compensate somewhat for that when we view things, probably using other clues that provide depth information, but not so much when we view photos (when depth clues are less available).

Fig 5. Green wedges highlight slope changes vs reference lines

The distorting effects of perspective are proportional to the angle of view, and thus can be reduced by narrowing the angle of view. Narrowing the angle of view also reduces the field of view, but field of view can be recovered by increasing distance (standing further back). Binoculars or a zoom lens compensate for distance. In practice, usually an outcrop is on a slope or in a canyon and therefore it is not easy to obtain distance, especially while maintaining a strike-parallel view. That often leaves us with considerable perspective distortion.

To explore the effects of perspective distortion, use the 'Lens' slider, which simulates the view through a 35mm zoom lens ranging from wide angle (24mm) to telephoto (250mm). The simulator automatically compensates for field of view by changing the 'distance', to give a feel for how much further back one might have to be. And as mentioned, many photos can only be taken while standing on the outcrop and hence are often wide angle. To make it easier to see the distortion, click 'Show dip reference lines', which display lines at the true strata dip; notice how in some views the strata dip changes over the field of view, due to perspective distortion (fig 5). Compare the dips in figure 5, which is a wide angle view, with those in the same outcrop in a telephoto view, figure 4.

Perspective distortion increases with the range of subject depth, and with angle of view. A plane perpendicular to view (with a range of depth of zero) is distortion-free with any angle of view. Photographing surfaces as closely to perpendicular as possible reduces perspective distortion. Getting distance (and using a zoom to compensate) helps deal with non-planar surfaces.

The angle of view vs focal length for some cameras is available here, at the bottom of some of the pages.

Fig 6. Lineations

Figure 6 to the right shows a block of crossbedding cut by an erosion surface. The patterns on the erosion surface (left face) are produced by the intersection of cross-strata with the erosion surface. If both the cross-strata features and the erosion surface are planar, their intersection produces lineations (lines). The dip and dip direction of any individual lineation (eg., at the blue arrow in the figure) can be directly measured (when the subject is a linear feature, its dip and dip direction are more specifically called plunge and trend). But to determine the dip of the associated cross-stratum requires more information, either the strike of the cross-strata, or a second exposure at a different outcrop surface angle. Both cases are consdiered below.

Fig 7. Spherical to cartesian coordinates

In figure 6, presumed-planar crossbed features are exposed on a slope (left face) and on a horizontal surface (top face). Thus we can obtain the plunge and trend (ie., dip and dip direction) of a lineation on the sloping left face (blue arrow), and can obtain the strike direction from one of the lines on the horizontal surface (green arrow). If the cross-strata are parallel, they need not be from the same stratum.

That provides two vectors: Strike vector **S**, and Lineation vector **L**.

The azimuth angle between S and L (ie., the angle between when projected on the horizontal plane) is simply the difference between their measured directions; call that θ. Let φ be the plunge of an outcrop line (eg., as at the blue arrow, fig 5):

θ = | strike - trend | | (ie., the difference in azimuths) |

φ = plunge | (ie., the dip of the lineation) |

Defining the vectors in Cartesian coordinates (fig 7), choosing the x-axis to align with strike vector S:

Strike vector S = | [1, 0, 0] |

Line vector L = | [cos(φ)cos(θ), cos(φ)sin(θ), sin(φ)] |

Vectors **S** and **L** are both in the slipface plane, and thus their vector cross product provides a vector normal (ie., perpendicular) to the slipface plane. A vector cross-product in matrix notation is the matrix determinant:

S X L = |
x | y | z |

1 | 0 | 0 | |

cos(φ)cos(θ) | cos(φ)sin(θ) | sin(φ) |

= [0, -sin(φ), cos(φ)sin(θ)]

That vector is, of course, in the y-z plane because the x-axis was chosen to align with **S**. So to find the slipface dip, all we need to do is find the angle of that vector in the y-z plane between it and the z-axis (because the vector is normal to the cross-stratum plane):

slipfaceDip = arctan | -sin(φ) | = arctan | -tan(φ) |

cos(φ)sin(θ) | sin(θ) |

Replacing the symbols with words:

slipfaceDip = arctan | tan(plunge) |

sin(diff btwn strike & trend azimuths) |

The cross-strata dip direction is simply perpendicular to the strike, which was given.

Given azimuth difference and plunge, the calculator to the right solves for dip, and shows the sensitivity of the result to perturbation.

If an outcrop doesn't expose the strike of any eligible cross-strata, maybe it is curved and exposes a cross-stratum (or parallel cross-strata) at different place, at different angles. If the cross-strata are planar, two different lineations from a cross-strata plane can provide enough information to find the dip of the cross-strata plane; if the cross-strata are parallel, the two lineations needn't come from the same stratum.

The derivation is similar to above; each lineation's plunge and trend defines a vector, each of which are in the plane of the cross-stratum, and so the cross product of the two provides a vector normal to the cross-stratum. Align one vector **L _{1}** with the x-z plane, and the other at angle θ away. Then the cross product is obtained by finding the matrix determinant:

L X _{1}L =_{2} |
x | y | z |

cos(φ_{1}) | 0 | sin(φ_{1}) | |

cos(φ_{2})cos(θ) | cos(φ_{2})sin(θ) | sin(φ_{2}) |

= [ | -sin(φ_{1})cos(φ_{2})sin(θ), |

sin(φ_{1})cos(φ_{2})cos(θ) - cos(φ_{1})sin(φ_{2}), | |

cos(φ_{1})cos(φ_{2})sin(θ)] |

In this case the cross product vector is not necessarily in the y-z plane (the y component is not zero) and therefore the denominator of the slope ratio involves both the x and y components (to find the length of the vector's projection in the x-y plane); that expression doesn't reduce to something as simple as in the strike+line case. But it is easy to calculate numerically, as performed by the calculator to the right.

The cross product vector is normal to the cross-strata plane, so the dip direction relative to the trend of the first lineation vector **L _{1}** is its projection onto the x-y plane, rotated by 180° (ie., arctan(-y/-x)):

arctan | -( sin(φ_{1})cos(φ_{2})cos(θ) -
cos(φ_{1})sin(φ_{2}) ) |

-( -sin(φ_{1})cos(φ_{2})sin(θ) ) |

which, using Wolfram's online 'simplify' utility, reduces to:

= arctan(csc(θ)(cot(φ_{1})tan(φ_{2}) - cos(θ)))

That's implemented in the tool to the right, with the dip direction displayed beneath the table of results. The dip direction angle, relative to the trend of **L _{1}**, is in compass degrees (ie., increases clockwise) by choice of the Cartesian axes (fig 7).

Fig 8. Idealized cross-stratified bedform; move your mouse over the columns below to see the associated angle of cut

Figure 8 shows another idealized bedform, viewed perpendicular to the direction of dune travel (i.e., perpendicular to strike), consisting of three sets of crossbedding. Columns below it display what would be visible on a planar erosion surface dipping in the same direction as the dune slipfaces. Move your mouse over each column to see the associated cut. There is a dramatic difference in apparent thickness of the strata that would be visible on the erosion surface, measured along the surface -- ranging from true thickness of 9.6 units to much higher (the column labels give the dip, sloping left or right, and the measured thickness 't' of a stratum).

The true thickness of an item (eg., stratum or set) is visible only when the erosion surface is perpendicular to that item. The cross-strata dip at 40° to the right, and thus the erosion surface dipping 50° to the left has a dihedral angle of 90° and therefore yields true thickness. Similarly, the erosion surface dipping at 90° is perpendicular to the horizontal set boundaries and therefore shows the sets (but not the dipping strata) at their true thickness. The apparent thickness of the strata and sets is larger at non-perpendicular exposure angles.

The following open source javascript packages, for which I'm grateful, were used to show crossbedding cut by an erosion surface: Three.js (3D graphics), csg.js (constructive solid geometry), threeCSG.js (a bridge between three.js and csg.js), jquery.nouislider.js (sliders), and jquery.js (a general-purpose library). GeoGebra was helpful for confirming rotations and vectors.

If you experience problems or have suggestions, please send a comment.

2014-Dec