How to calculate an apparent dip from a real dip (and viceversa) using orthographic projection and trigonometry

Any student of geology in any university in the world learn during its degree the relationship between the real dip and the infinite apparent dips that a plane contains. Most of students learn how to calculate a real dip from a couple of apparent dips or, inversely, how to work out an apparent dip given the real dip and another direction using the stereonet. 

Stereographic projection provides an awesome graphic method for these calculation, which is very useful if one has to do a bunch of calculation, but it is not very useful if we have to deal with a large collection of data, or if we need to have a high degree of precision. Then, it is time for orthographic projections and the always useful trigonometry. 

Let's imagine that we have a known plane, and we need to calculate an apparent dip. We know the dip angle and the dip direction or strike, and obviously we know the direction along we want to know the apparent dip. In this context, 

δ = real dip. Note that the real dip is always measured along the maximum slope direction for a plane. No apparent dip can be larger than the real dip. 

α = apparent dip. This is the dip measure along a line which is not the maximum slope direction.

β = angle between the strike direction of the plane and the apparent dip direction.

You could think that it is difficult, but it is actually quite easy. The "trick" lies on relating the three triangles involved in the diagram (one containing a and b, another containing c and b, and another containing a and c). (Note that c is the hypotenuse of the horizontal triangle)

 The following trigonometric relations are quite straight and don't need much explanation:

(1)           sin β = a/c;  a = c ∙ sin β

(2)           tan δ = b/a; b = a ∙ tan δ

(3)           tan α = b/c; b = c ∙ tan α

(4)           b = a ∙ tan δ

(5)           b = c ∙ tan α

(6)           a = c ∙ sin β

Clear so far? Now, if you equal (4) and (5), and substitute a by (6),

(7)           a ∙ tan δ = c ∙ tan α

(8)           c ∙ sin β ∙ tan δ = c ∙ tan α

(9)           sin β ∙ tan δ = tan α

what you obtain is a direct relation between  α and δ. If you want to know the real dip from an apparent dip, use (11). If you want to calculate the apparent dip from the real dip, then use (10)

(10)          α = arctan (sin β ∙ tan δ)

(11)          δ =  arctan (tan α / sin β)

Easy, isn't it?

Why you would need to use that? Well, for example, I need it sometimes; I work interpreting satellite images, focusing on structural geology. When I measure fracture lengths on a plane, I cannot really measure their length: What I measure is an "apparent length". That means, I measure the projection of a line on a horizontal plane. For example, a 100 m fracture on a plane dipping 80 degrees will look very short if the direction of that fracture is the dip direction of the plane, but it will look as 100 m if the fracture is oriented along the strike. Any direction in between, will be variable. If it is variable, how can we correct it? knowing the apparent dip in that direction.

This method provides a way of correcting this distorsion, simply using any spreadsheet. You know the length of every single fracture, and the length you have measured. You also know the real dip of the plane (well, I can measure it on the DEM!), the strike, and that is all you need to know. But this will be another explanation, coming soon :-)

Feel free to make any comments, or perhaps any correction of suggestion. Hopefully this has been useful for you. 

Geo-vandalism in Bartlett Wash, Utah

Dr. Bruce Trudgill (assistant professor at Colorado School of Mines) has reported in the Geotectonics email list an act of geo-vandalism in the Bartlett Wash outcrop, in southeast Utah (USA).

Quoting Bruce,

Many of you may have visited this spectacular location on university or industry field trips, or for your own research purposes. The location of the exposure is on a splay off the main Moab Fault, and it illustrates many aspects of brittle deformation and fluid flow, as well as some un-paralled exposures of aeolian dune sets in the Slickrock member of the Entrada Formation. It's a truly world-class field location and has been used in a number of publications and texts, including the following figure in Haakon Fossen's structural geology textbook.

Figure 8.11 in Structural Geology by Haakon Fossen (Cambridge University Press)

One of the key aspects of this location is the 100% exposure of deformation bands in the footwall of the fault and their relationship to fluid flow. Students can measure and plot deformation band density in the footwall of the fault and it's a great location to discuss their influence of fluid migration. Anyway, this (formally) pristine outcrop is now missing a few of the deformation bands in Haakon's photo due to some mindless geo-vandalism (see below)

Photo taken by Roy Luck on September 29th 2011.......note the rock powder spread around the outcrop showing evidence of very recent cutting

[end of quotation]

The exact coordinates of the affected area are 38° 43' 00.09" N  109° 47' 17.85" W:


View Larger Map

This act of vandalism took place between 25th and 29th of September 2011. The following picture portraits Bruce on the 25th, early evening:

Bruce Trudgill in the intact outrop. Red circle shows the area later on vandalised by unknown people.

If you know who has done that, you should contact Bruce (http://geology.mines.edu/faculty/btrudgil/) or Becky Doolittle (rdoolitt at blm (d.o.t.) gov) at the Bureau of Land Management. Vandalism is an act of crime.

Why people do that? Reasons (excuses?) are not short. Some people do it for an economic interest: Nice primary or secondary structures, fossils, minerals, etc,  are aesthetically attractive, and they can be sold for good money. Researchers need to take samples. But things can be done right, or wrong. Cutting off a cube of rock with an electric saw in an outstanding outcrop, in such a visible place, is plainly wrong.

If the responsible guys of this act of vandalism are researchers, sooner or later others will know. They will be the ones who spoilt Bartlett Wash. For future researchers, think twice what you do on the field, and think how would you like to find an outcrop if you would be the next person to arrive. You are scientist, full stop. Preservation of nature has to be your main objective, always. 


Thanks Bruce for raising the voice about this sad issue. Many of us have never been there, or even near, but that doesn't mean we don't feel something has to be done. Thanks also for the pictures and the permission for the quotations!

Thursday video: The Magic Toilet Paper

Thursday video will be a fixed section in this blog where we will link toYouTube clips (or other sites, of course!) related with tectonics and structural geology. If you want some video displayed here, please feel free to let me know and I will do it.

Let's start with this video from the Structural geology, Tectonics and Geomechanics Research Group of the University of Aachen (Germany). It shows a toilet paper roll being compressed, and behaving like a multilayer. It develops some chevron folds and kink bands, and what it is very interesting, you can do that at home with some DIY skills...

Ladies and gentlemen... "The Magic Toilet Paper"!

You can subscribe to their YouTube channel, or follow them in Twitter. Anyway, you can be sure they will be regularly featured here, as they have been doing for some time very nice and cool stuff.

USGS make available topographic maps online

USGS has just announced that their historic collection of topographic US maps goes digital. More than 161,000 maps have been digitased and made available online here: http://nationalmap.gov/historical/

If you are a student in the US, that should be good news for you, as you won't have to spend a dollar in maps for working in your own doing geological mapping. Which you should do, as practise makes perfection! ;-)

Extensional grabens in our shrinking Moon

The largest of the observed grabens with the
 Narrow Angle Camera (NAC) of the LRO spacecraft. 
It is about 500 m wide, and nearly 20 m deep. 
(Credit: NASA/Goddard /
 Arizona State University/Smithsonian Institution)

Who has not read a thousand times that our Moon is a death body without any activity? Well, it turns to be the opposite: The Moon has been active at least until very recent times, and very likely this activity continues now.

Lunar Reconnaissance Orbiter (LRO) spacecraft has obtained images of the surface of our satellite where extensional faults are clearly visible. NASA estimates that this faults are not older than 150 million years, which is pretty young for what we expected to be the lunar tectonics.

Two years ago, the Lunar Reconnaissance Orbiter mission also detected other young geological features, which where interpretd as compressional uplifts. It was said, at that time, that the Moon is shriking. But the identification of extensional faults shows that this shrinking (due to thermal contractions) is not homogeneus everywhere.


But why not listening directly to Tom Watters from the Smithsonian Institution? (You can also active the subtitles!):

Diagram of formation of a graben with two bounding faults (Credit: Arizona State University/Smithsonian Institution)

Read more here: http://www.nasa.gov/mission_pages/LRO/news/lunar-graben.html

The origins of plate tectonics, with Dan McKenzie and Fred Vine

The Geological Society of London opened a few months ago a channel in YouTube with the aim of reaching a larger audience through divulgation videos.

Yesterday, The Geological Society uploaded a very interesting inteview to Dan McKenzie and Fred Vine, where they discuss the early history of development of the theory of plate tectonics. The clip is taken from 'Dan McKenzie and friends: highlights from the Bullard labs'. [Copyright: 2011 Cambridge University (Depart. of Earth Sciences)].

Enjoy it!

http://www.youtube.com/watch?v=MQY-aaXz2c4

Structural Geology Lab Manual by David Allison

A structural geology laboratory manual comes always handy to anyone dealing with maps, cross-sections or stereograhic nets. Even those experienced geologists who have left their university days left behind the mist of time need to have one. Many techniques are learnt, but soon leave room in our room for other issues that may be even more important (e.g. telephone number of the canteen waitress, local pubs near an outcrop, and so forth...)

There are methods in structural geology that if one doesn't use them regularly, soon become rusty and we simply need to refresh them from time to time: how to rotate lines and planes with a stereonet, how to solve a three-point problem, etc.

Today I'd like to present you an excellent structural geology lab manual, written by David T. Allison, an associate professor of geology of the Department of Geology and Geography of the University of South Alabama:

http://www.usouthal.edu/geography/allison/GY403/StructuralGeologyLabManual.pdf

The manual contains explanations and exercises on attitude measurements, true and apparent dips, three-point problems, stereographic projections, rotations with the stereonet, stereograms, geologic mapping and cross-section construction, thickness and outcrop problems ans statistical techniques.
It is written in a very approachable style, and completed with good figures that will help anyone to understand and practise the foundations of our profession or studies.

The text makes reference to several spreadsheets that you can find in the homepage of the author:

http://www.usouthal.edu/geography/allison/ 

 I hope you find it useful. Comments are welcomed, as usual!