Find Trend And Plunge, Given Pitch

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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It is sometimes useful to characterize the orientation of a line by referring to its direction in a dipping plane. For example, you may have ripple marks on a bed or slickensides on a fault, and it may be difficult to determine their trend and plunge accurately. On the other hand, it is very important that these lines are contained within some particular plane (the bed and the fault, respectively). In cases like these, the pitch measurement is sometimes used.

Pitch is defined as the angle between some line in a plane and a horizontal line, measured in the plane. A line can have the same pitch in two directions, so it is important to define directions precisely.

             _______________________________________
Pitch 60    /             60 ( + ) 60              /
degrees NE /                 +  +                 / Pitch 60
          /                +     +               /  degrees SW
         /               +        +             /  
        /              +           +           /  
       /             +              +         / 
      /            +                 +       / 
     /______________________________________/

    <-------n33e (033) S33W (213)------->

          Fault:  Strike N33E, dip 56NW

We can solve the problem easily if we note that structure contours on a map are foreshortened views of the real thing. If the plane dips with an angle D, the mapped contours are compressed by a factor of cos D


                   Map Spacing S
 -------------A+-----------------+-----
                  + ) D          |
                     +           |       S = T Cos D
            True        +        |
            Spacing   T    +     |       T = S/Cos D 
            in Plane          +  | 
                                 +B

You can determine the true spacing of the contours by calculation or by measuring distance AB in the cross-section.


MAPPED CONTOURS AND MAP PROJECTION OF PITCHING LINE  
------------------A----------- ___ ___-----------+---------------
               +  |               | 60           | S
------------B-----|------------------------------+---------------
         +        |
------C-----------|----------------------------------------------

TRUE VIEW OF CONTOURS AND PITCHING LINE IN DIPPING PLANE 
                  
------------------A------------------------------+---------------
          Pitch ( |                              |
               +  |                              | S/Cos 60
                  |                              |
------------B-----|------------------------------+---------------
                  |
         +        |     
                  |
------C-----------|----------------------------------------------

Example

1. Given the fault shown and slickensides with the observed pitch, find their trend and plunge. 2. Construct structure contours for the plane. Find the true spacing by cross-section or trigonometry and construct a second set of contours with true spacing. 3. On the true set of contours, construct the pitching line as it appears on the dipping plane. Project distances along the contours back to the map view. 4. On the map view, construct the map trace of the line. Find its trend and find the plunge by trigonometry or drawing a cross-section.

Mathematical Note


MAPPED CONTOURS AND MAP PROJECTION OF PITCHING LINE  
------------X-----A----------- ___ ___-----------+---------------
               +  |               | D            | S
------------B-----|------------------------------+---------------
         +        |
------C-----------|----------------------------------------------

TRUE VIEW OF CONTOURS AND PITCHING LINE IN DIPPING PLANE 
                  
------------X-----A------------------------------+---------------
          Pitch ( |                              |
               +  |                              | S/Cos D
                  |                              |
------------B'----|------------------------------+---------------
                  |
         +        |     
                  |
------C'----------|----------------------------------------------

We know the pitch and the dip. What we want to find is the trend and plunge. The trend is just the strike of the bed (given by the structure contours) plus or minus angle XAB in the top diagram. It will be plus or minus depending on which way the pitch is measured relative to the strike.

We have:

Tan XAB = XB/AX
Tan(Pitch) = XB'/AX. Thus:
Tan XAB/Tan(Pitch) = XB/XB' = S/(S/Cos D) = Cos D
Therefore, Tan XAB = Tan(Pitch)Cos D

The plunge (P) is found from the diagrams below:

Cross-Section Down Dip  Cross-Section Along Line   True View of Plane

---X+---------B---     --A+------------------B     ------X------A-----
      +) D    |              + )P            |           Pitch (        
        +     | Contour          +           |           |    +       
          +   | Interval             +       |           |   +   
            + |                          +   |           |  +    
              +B'                            +           | +
                                             B'    ------+B'----------
                                                   B' is below point B

We have:

BB' = XB' Sin D
BB' = AB Tan P
XB' = AB Tan(Pitch)
The first two equations give us AB Tan P = XB'Sin D
Substituting XB' from the third equation gives us AB Tan P = AB Tan(Pitch)Sin D or
Tan P = Tan(Pitch)Sin D

Final Result

If X is the angle between the trend and the strike, P is the plunge, D is the dip of the layer, then

Tan X = Tan(Pitch)Cos D
Tan P = Tan(Pitch)Sin D

Special Cases

If Pitch = 0, trend=strike, plunge = 0
If Pitch = 90, trend=strike plus or minus 90, plunge = dip

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Created 5 January 1999, Last Update 20 January 1999
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