Rules of thumb:

• Strata always appear thicker than they really are (except when cut perpendicular to the dip)
• Dips always appear shallower than they really are (except when cut perpendicular to the strike)

True Dip

Fig 1. Direct measurement

Erosion sometimes exposes a portion of an ancient slipface such that its dip can be measured directly, but more often all that is available are cross-strata traces on an outcrop surface. In that case, each stratum, which is a line (or curve) appearing on the outcrop surface, plunges at an angle less than the true dip of the slipface plane containing it. To find the true dip, we need to use other information.

The slipface in diagram below (figure 3) slopes down toward us. Any plane (eg., an outcrop surface) cutting that slipface plane will create a stratum line, and any horizontal plane cutting the slipface plane will create a line of strike. The true dip of the slipface plane is the angle of dip measured in the plane perpendicular to its line of strike.

Fig 2. Slipface strata on outcrop surface

Usually an outcrop surface is not vertical or perpendicular to an ancient slipface within the sandstone. Most commonly we see intersections of slipface features (eg., pinstripes or grainflow laminae) exposed as lines on an erosion surface (assuming the simple case where the entities are all planar). We can measure the plunge and trend of any individual line on that surface, and if we're lucky (if it's exposed), we can measure the strike of the slipface plane. From that information, we can calculate the true dip of the slipface plane, as derived below.

Fig 3. True and apparent dip (adapted from Martel)

Given the plunge and trend of a line created by the intersection of a slipface with an erosion plane, and the dip and strike of that erosion plane, we can find the true dip of the slipface plane as follows:

The left table below gives TrueDip_slipface for values of Plunge_stratum (between 0° and 45°), and the angle between the trend of an exposed lamina and the strike of the slipface plane. The right table is the difference between the plunge and slipface true dip -- this highlights that the difference is insignificant if the erosion plane is close to perpendicular to the slipface.

Determining the plunge and trend of laminae on the erosion surface is straightforward, but finding the strike of the slipface plane can be difficult, though often there is fortunate exposure somewhere nearby. Of course slipfaces aren't necessarily planar.

True Thickness

Fig 4. Apparent thickness of layer is 1.5cm

In figure 4 to the right, the ruler says the avalanche layer between the arrows is 1.5cm thick -- is that apparent thickness the true thickness of the layer, as measured perpendicular to its bounding surfaces?

The true thickness of the layer is always less than (or, at best, equal to) the apparent (measured) thickness, depending on the angle at which the outcrop surface cuts the layer.

In the general case, where an outcrop surface may be at an oblique angle to a slipface layer, the apparent thickness of the layer depends on the dihedral angle between the two planes. This can be calculated using vectors: If n_outcrop and n_slipface are unit vectors normal to their planes, then the true thickness t_true of a slipface layer is given by t_apparent * (n_outcrop · n_slipface), where 'n_outcrop · n_slipface' is the dot product of the two unit vectors, a scalar less than or equal to 1. We can perform this calculation by converting outcrop measurements into cartesian coordinates.

Fig 5. Spherical to cartesian

In figure 5, the dip direction θ (azimuth) and dip Φ to point P can be represented by a unit vector P as follows (bold font indicates a unit vector):

Thus:

Finding n_slipface is equally straightforward if we can directly measure the true dip and dip direction of the slipface within the outcrop. If not, we may be able to obtain n_slipface from two lines in the slipface plane -- the trace of a slipface stratum on the outcrop, and the direction of dip of the slipface (ie., perpendicular to its strike). n_slipface can then be found since the cross-product of two vectors in a plane results in a vector normal to that plane.

A unit vector in the direction of a slipface stratum is:

A horizontal (ie., dip = 0) unit vector in the direction of the slipface dip is:

The cross-product of these yields n_slipface:

n_slipface = n_stratum X n_{slipfaceHorDir}

and finally we can find:

 t_true = t_apparent * (n_outcrop · n_slipface) 

Here's an Excel spreadsheet to do these calculations.

If the slipface and outcrop surface dip in roughly the same plane (ie., their lines of strike are parallel), the above simplifies to:

 t_true = t_apparent * sin(| Dip_outcrop - Dip_slipface |)  [when slipface and outcrop dip in same plane]

If the outcrop surface is close to vertical (ie., dip ~= 90°):

Consider the idealized bedform in figure 6 to the right and how it would appear if cut by an erosion surface with a parallel line of strike but with a dip at various angles. Columns below the 'bedform' display what the surface would look like at the given dip angle. Notice the dramatic difference in apparent thickness of the layers. Move your mouse over the columns to see the bedform cut by erosion surfaces at the given angle. The column labels give the dip (sloping left or right) and the measured thickness of a layer. The strata dip 40° right, and thus an erosion surface dipping 50° left has a dihedral angle of 90° and therefore yields apparent = true thickness.

Of course not just the thickness of individual stratum are affected; so is the apparent thickness of the cross-strata set.

References

Stephen J. Martel, Structural Geology Lecture Notes, Geology and Geophysics, University of Hawai'i

Above, a simulated block of cross-bedded sandstone. Only in profile (dip section, where the cut is a vertical plane parallel to the dip direction) do the strata appear in their true thickness. Click on a face to rotate it into face-on position, and click-drag to rotate the block. Credits: Eric Lin for the rotating cube Flash code.