Appendix 1: Calculating fault-perpendicular elliptical plume half length
If a plume margin is assumed to be elliptical, then k (half the length of the plume
perpendicular to a fault) can be computed from the area of the plume, Ap, the aspect ratio
or eccentricity of the plume (the length or semimajor axis divided by the width or
semiminor axis), R, and the acute angle between the fault and the plume axis,  f , using
the parametric approach.
The semimajor axis, a, and semiminor axis, b, can be computed from Ap and R.
a
b
(A-1)
a  Rb
(A-2)
A p  ab
(A-3)
Ap  Rb 2
(A-4)
R
So,
The area of an ellipse is given by
Substituting Equation A-2,
Solving for b,
b
Ap
R
(A-5)
With b, a can be computed from Equation A-2.
Figure A-1 shows the variables for derivation of the general parametric equations for the
elliptical angle,  , the tangential angle,  , the Cartesian coordinates, x and y, and the
length of the radius, r, for points on the ellipse,
A-1
Figure A-1. Diagram of variables for the parametric definition of an ellipse.
The parametric equation for  can be derived as follows.
tan  
PQ
OP
(A-6)
From compression of a circle to create an ellipse,
PQ 
b
PR
a
PR  OP tan t
(A-7)
(A-8)
Substituting Equation A-8 into A-7 gives,
PQ 
b
OP tan t
a
(A-9)
Substituting Equation A-9 into Equation A-6 and canceling terms yields,
tan  
b
tan t
a
A-2
(A-10)
Solving for  ,
b
a
  tan 1 tan t
(A-11)
Similarly, the parametric equation for  can be derived as follows.
PQ
PS
(A-12)
b
PR
a
(A-7)
tan  
Again,
PQ 
But for  ,
PR  PS tan f
(A-13)
From complementary angles,
f 

2
t
(A-14)
Substituting in Equation A-13,
PR  PS tan(

2
 t)
(A-15)
But,
tan(

2
 t )  cot t
(A-16)
Substituting into Equation A-15,
PR  PS cot t
(A-17)
Substituting into Equation A-7,
PQ 
b
PS cot t
a
Substituting into Equation A-12 and canceling terms,
A-3
(A-18)
tan  
b
cot t
a
(A-19)
Solving for  ,
b
a
  tan 1 cot t
(A-20)
Deriving the x coordinate for a point on the ellipse at t,
x  OP
(A-21)
OP  a cos t
(A-22)
x  a cos t
(A-23)
So,
Deriving the y coordinate for a point on the ellipse at t,
y  PQ
(A-24)
b
PR
a
(A-25)
PR  a sin t
(A-26)
PQ 
Substituting Equation into A-25 and canceling terms,
PQ  b sin t
(A-27)
y  b sin t
(A-28)
So,
The length of the elliptical radius, r, at t is,
r
x2  y2
(A-29)
Substituting from Equations A-23 and A-28
r  a 2 cos 2 t  b 2 sin 2 t
(A-30)
The plume dimension perpendicular to a fault can be computed by finding the length of a
normal from the center of the ellipse to a tangent parallel to the fault. Figure A-2 shows
the variables for this derivation.
A-4
A-5
A-6
A-7
Figure A-2. Diagram for the derivation of the ellipse dimension perpendicular to a fault.
The parametric value at a point on the ellipse with a tangent parallel to the fault, tt, can be
found by setting  equal to  in Equation A-12, and solving for tt. From Equation A-20,
  tan 1 cot t t
b
a
(A-31)
a
tan 
b
(A-32)
Solving for tt,
t t  cot 1
The normal distance between the center of the ellipse and the fault parallel tangent, k, is
k  rt cos 
(A-33)
where rt is the length of the elliptical radius to the point of fault parallel tangency, and 
is the angle between the normal to the fault,  p , and the angle of the tangent radius,  t .
The value  is,
   p  t
A-8
(A-34)
From complementary angles,
p 

2

(A-35)
The value of  t can be found from Equation A-11 using tt. Substituting this and Equation
A-35 into Equation A-34 gives,


2
   tan 1
b
tan t t
a
(A-36)
Substituting the value of  from Equation A-36, and the value of rt from using tt in
Equation A-30, into Equation A-33 gives the value of k.
A-9