Tuesday, 1 May 2012

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. 

11 comments:

Rowan Cockett said...

Nice write up!

I made an interactive little tutorial on Apparent Dip a few weeks ago. Where you can play around with a cross-section through a dipping layer.

http://app.visiblegeology.com/apparentDip.html

You have a much better explanation and derivation, but it is always nice to play around with this stuff in 3D to get a feel for it!

Anonymous said...

Rowan, the apparent dip has a very faulty and confusing 3D display, strike direction and cross section direction weren't correct

mua thuốc tốt said...

Collagen có nhiều tác dụng , vậy collagen co tac dung gi ? tác dụng chính của collagen là giúp đẹp da và ngăn ngừa lão hóa da . cach bo sung collagen hiệu quả như ăn các loại rau xanh , thực phẩm nhiều omega-3 ,... Giảm cân có nhiều cách , trong đó dùng thực đơn giảm cân cấp tốc trong 1 tuần rất có hiệu quả . Mất ngủ xảy ra ở hầu hết các lứa tuổi , mất ngủ có ảnh hưởng đến thai nhi không ? Mất ngủ có ảnh hưởng khá lớn đến thai nhi . Bổ não và tăng cường trí nhớ có nhiều cách , trong đó thức ăn bổ não tăng cường trí nhớ được nhiều người áp dụng và rất hiệu quả mỗi ngày . Tai biến mạch máu não rất nguy hiểm , nên việc tìm hiểu về cách phòng ngừa tai biến mạch máu não là rất quan trọng .

Cindy Dy said...


I am very happy to locate your website. I just wanted to thank you for the time you spent on this great article. I definitely enjoyed reading it and I have you bookmarked to check out new stuff you post.


Yong
www.gofastek.com

Aydin Ahmadli said...

this was the one of the most excellent explanations of apparent dip amount! Thank you much!..

sarah lee said...

I really enjoyed reading your article. I found this as an informative and interesting post, so i think it is very useful and knowledgeable. I would like to thank you for the effort you have made in writing this article.


edupdf.org

packers movers said...

movers and packers Ahmedabad to Lucknow charges
movers and packers surat to chandigarh charges
movers and packers rajkot to Lucknow charges
movers and packers Rajkot to chandigarh charges
movers and packers Lucknow to Amritsar charges
movers and packers chandigarh to rajkot charges
movers and packers Amritsar to surat charges
movers and packers Ludhiana to Ahmedabad charges
movers and packers panchkula to Ludhiana charges
movers and packers mohali to varanasi charges
movers and packers Jalandhar to agra charges
movers and packers varanasi to Jalandhar charges
movers and packers allahabad to varanasi charges
movers and packers Kanpur to Bangalore charges
movers and packers agra to Bangalore charges
packers and movers in Bangalore to agra charges
packers and movers chandigarh charges
packers and movers lucknow charges
packers and movers Vadodara charges
packers and movers Ahmedabad charges
packers and movers surat charges
packers and movers Rajkot charges

packers movers said...

movers and packers Kolkata to delhi charges
movers and packers delhi to Mumbai charges
movers and packers Mumbai to Nagpur charges
movers and packers pune to Faridabad charges
movers and packers Faridabad to agar charges
movers and packers indore to agar charges
movers and packers chandigarh to trichy charges
movers and packers Chennai to Bangalore charges
movers and packers Bangalore to Chennai charges
packers and movers lucknow to Noida charges
movers and packers Noida to lucknow charges
movers and packers Ghaziabad to chandigarh charges
movers and packers Gurgaon to Ghaziabad charges
movers and packers movers thane to Allahabad charges
movers and packers navi Mumbai to Hyderabad charges
movers and packers Hyderabad to Bhubaneswar charges
movers and packers Bhubaneswar to lucknow charges
movers and packers Raipur to Bhubaneswar charges
movers and packers patna to Raipur charges

packers movers said...

packers and movers Chennai to Noida
packers and movers Noida to indore
packers and movers Ghaziabad to indore
packers and movers indore to Kolkata
packers and movers Gurgaon to Kolkata
packers and movers Kolkata to Mumbai
packers and movers Mumbai to Pune
packers and movers Pune to Hyderabad
packers and movers Hyderabad to Bangalore
packers and movers Bangalore to Faridabad
packers and movers Faridabad to navi Mumbai
packers and movers Delhi to Kanpur
packers and movers lucknow to Delhi


packers movers said...

car transport from chandigarh to Vadodara cost
car transport from lucknow to Vadodara cost
car transport from Kolkata to Vadodara cost
car transport from Hyderabad to Vadodara cost
car transport from Bangalore to Vadodara cost
car transport from pune to Vadodara
car transport from Mumbai to Vadodara cost
car transport from Chennai to Vadodara cost
car transport from delhi to Vadodara
car transport from Gurgaon to Vadodara cost
car transport from Noida to Vadodara
car transport from Faridabad to Vadodara cost
car transport from Ahmedabad to Vadodara cost
car transport from navi Mumbai to Vadodara cost
car transport from thane to Vadodara cost
car transport from jammu to Vadodara cost
car transport from surat to Vadodara
car transport from Vadodara to Vadodara cost
car transport from Nellore to Vadodara cost
car transport from panchkula to Vadodara cost
car transport from amritsar to Vadodara cost
car transport from Bhavnagar to Vadodara cost
car transport from rajkot to Vadodara cost
car transport from mohali to Vadodara cost
car transport from nagpur to Vadodara
car transport from nashik to Vadodara cost
car transport from Kalyan dombivali to Vadodara cost
car transport from vasai virar to Vadodara cost

packers movers said...

car transport from bhopal to Vadodara cost
car transport from indore to Vadodara cost
car transport from kanpur to Vadodara cost
car transport from varanasi to Vadodara cost
car transport from allahabad to Vadodara cost
car transport from Coimbatore to Vadodara cost
car transport from Trivandrum to Vadodara cost
car transport from cochin to Vadodara cost
car transport from mysore to Vadodara cost
car transport from kochi to Vadodara cost
car transport from visakhapatnam to Vadodara cost
car transport from vijayawada to Vadodara cost
car transport from trichy to Vadodara cost
car transport from goa to Vadodara cost
car transport from jaipur to Vadodara cost
car transport from patna to Vadodara cost
car transport from ranchi to Vadodara cost
car transport from raipur to Vadodara cost
car transport from guwahati to Vadodara cost
car transport from jalandhar to Vadodara cost
car transport from jamshedpur to Vadodara cost
car transport from dehradun to Vadodara cost
car transport from jamnagar to Vadodara cost
car transport from gandhinagar to Vadodara cost
car transport from ludhiana to Vadodara cost
car transport from shimla to Vadodara cost
car transport from bhubaneswar to Vadodara cost