Environmental and Exploration Geophysics II
Gravity Methods (VII) more wrap up
tom.h.wilson
tom.wilson@mail.wvu.edu
Department of Geology and Geography
West Virginia University
Morgantown, WV
Tom Wilson, Department of Geology and Geography
0.4
Bouguer Anomaly (mGals)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-1500
-1000
-500
0
500
1000
1500
Distance from peak (m)
X
z
r
Sphere with
radius R and
density 
Tom Wilson, Department of Geology and Geography
g vert




3
G  (4 / 3 R ) 
1


3 
 2
2
Z2


x



1
 
 z2
 

X
gvert
z
r
Sphere with
radius R and
density 
Tom Wilson, Department of Geology and Geography
Diagnostic position X1/2
gv
1
1
 
3/ 2
2
g max 2  x

1/ 2
 2  1
 z

x½ is referred to as the
diagnostic position, 1/x1/2 is
referred to as the depth
index multiplier
Tom Wilson, Department of Geology and Geography
We solve for
x1/2/z and find
that x1/2/z =
0.766.
The values at several points along the X/Z axis can be
calculated to define the shape of the anomaly
gv
g max
3/4
1/2
1/4
Tom Wilson, Department of Geology and Geography
The other day we developed these tables of diagnostic
positions and depth index multipliers
Diagnostic Position
(g/gmax)
3/4 max
2/3 max
1/2 max
1/3 max
1/4 max
Depth Index Multiplier
1/0.46 = 2.17
1/0.56 = 1.79
1/0.77 = 1.305
1/1.04 = 0.96
1/1.24 = 0.81
Note that regardless of which diagnostic position you
use, you should get the same value of Z. Each depth
index multiplier converts a specific reference X location
distance to depth.
Z  (depth index multiplier) times X at the diagnostic position
Tom Wilson, Department of Geology and Geography
Check it out ….
Depth index multiplier
for X1/2 is 1.305
Depth index multiplier
for X3/4 is 2.17
What depth do you get?
X1/2 X3/4
m Wilson, Department of Geology and Geography
Once you figure out Z - Solve for R or 
1/ 3
 g max Z 2 
R

(4
/
3

)
G




(feet)
G  (4 / 3 R 3 )
g max 
Z2
R3
 0.02793 2  for meters
Z
R3
 0.00852 2  for feet
Z
1/ 3
 g max Z 2 
R

0.00852




g max Z 2
 
0.00852 R 3
Tom Wilson, Department of Geology and Geography
(feet)
(feet)
gmax Z 2
 
(4 / 3 )GR3
(feet)
These constants (i.e.
0.02793 or 0.00852)
assume that depths
and radii are in the
specified units (feet
or meters), and that
density is always in
gm/cm3.
X
z
Cylinder
with radius R
and density 
R
Tom Wilson, Department of Geology and Geography
X
z
r
At surface
distance x away
from a point
directly over
the cylinder
Tom Wilson, Department of Geology and Geography
Results for Horizontal Cylinder
gcyl


2 G  R 2  1 

 x 2  1
Z
 z 2 
g cyl
Tom Wilson, Department of Geology and Geography
and
g max


1 
 g max  2
x


1
2
 z

2 G  R 2

Z
We can ask the same kinds of questions we
asked regarding the sphere. For example,
Where does
g cyl
g max

1
2
1
1
 2
2 x 2 1
z
x2
z
2 1  2
x2
z
2
1
x
1
z
x1  z
2
This tells us that the anomaly falls to ½ its maximum
value at a distance from the anomaly peak equal to the
depth to the center of the horizontal cylinder
Tom Wilson, Department of Geology and Geography
X3/4X2/3
Locate the points along the
X/Z Axis where the
normalized curve falls to
diagnostic values - 1/4, 1/2,
etc.
X1/4 X1/3
0.58
0.71
The depth index multiplier is
just the reciprocal of the
value at X/Z at the
diagnostic position.
X times the depth index
multiplier yields Z
0.58
0.71
1.0
1.42
Tom Wilson, Department of Geology and Geography
X1/2
1.74
Z=X1/2
Simple relationships and formula
for the horizontal cylinder
Diagnostic Position
3/4 max
2/3 max
1/2 max
1/3 max
1/4 max
G 2R 2 
g max 
Z
R2
 0.0419  for meters
Z
R2
 0.01277  for feet
Z
1/ 2
 g max Z 
 (feet)
R  
0
.
01277




g max Z
 
(feet)
0.01277R 2
Tom Wilson, Department of Geology and Geography
Depth Index Multiplier
1/0.58 = 1.72
1/0.71 = 1.41
1/1= 1
1/1.42 = 0.7
1/1.74 = 0.57
With Z, you can then speculate
on the density contrast or radius
of the object in question.
Again, note that these constants
(i.e. 0.02793) assume that depths
and radii are in the specified units
(feet or meters), and that density is
always in gm/cm3.
Horizontal Cylinder
Just as was the case for the sphere, objects which have a
cylindrical distribution of density contrast all produce variations
in gravitational acceleration that are identical in shape and differ
only in magnitude and spatial extent.
When these curves are normalized and plotted as a function
of X/Z they all have the same shape.
It is that attribute of the cylinder and the sphere which
allows us to determine their depth and speculate about the
other parameters such as their density contrast and radius.
Tom Wilson, Department of Geology and Geography
Assume the anomaly below is produced by long
horizontal tunnel – What is the depth to the tunnel?
What are the depth
index multipliers?
X1/2
Tom Wilson, Department of Geology and Geography
X3/4
What is the approximate shape of the object
producing the anomaly below?
Diagnostic
positions
X3/4 = 0.95
X2/3 = 1.15
X1/2 = 1.6
X1/3 = 2.1
X1/4 = 2.5
Multipliers
Sphere
2.17
1.79
1.305
0.96
0.81
ZSphere
2.06
2.06
2.09
2.02
2.03
Multipliers
Cylinder
1.72
1.41
1
0.7
0.57
ZCylinder
1.63
1.62
1.6
1.47
1.43
Which estimate of Z seems to be more reliable? Compute the range.
You could also compare standard deviations.
Which model - sphere or cylinder - yields the smaller range or standard deviation?
Tom Wilson, Department of Geology and Geography
Estimating other properties of the buried object
To determine the radius of this object, we can use the formulas we
developed earlier. For example, if we found that the anomaly was
best explained by a spherical distribution of density contrast, then
we could use the following formulas which have been modified to
yield answer’s in kilofeet, where Z is in kilofeet, and  is in gm/cm3.
1/ 3
2
g
Z
R   max 
 8.52  


 
g maxZ 2
8.52 R
Tom Wilson, Department of Geology and Geography
3
(kilofeet)
(kilofeet)
For the remainder of the class consider the inclass activity
Tom Wilson, Department of Geology and Geography
sphere
10
cylinder
10
Vertical Cylinder
Diagnostic Position
3/4 max
2/3 max
1/2 max
1/3 max
1/4 max
Depth Index Multiplier
1/0.86 = 1.16
1/1.1 = 0.91
1/1.72= 0.58
1/2.76 = 0.36
1/3.72 = 0.27
R2
 0.01886  for meters
Z1
R2
 0.000575  for feet
Z1
1/ 2


1
1


g  G  R 2 

1/ 2 
2
2
2 1/ 2
2

 z  x 

z  L  x

 

Tom Wilson, Department of Geology and Geography
 g max Z1 
R

 0.000575 
g max Z1
 
0.000575R 2
(feet)
(feet)
For a given anomaly certain simple geometries
can be assumed and tested
Horizontal cylinder or
vertical dyke
Sphere or vertical cylinder
Tom Wilson, Department of Geology and Geography
How about the anomaly below?
Half plate or faulted plate
Tom Wilson, Department of Geology and Geography
Problem 6.5
6.5 What is the radius of the smallest equidimensional void
(e.g. chamber in a cave) that can be detected by a gravity
survey for which the Bouguer gravity values have an
accuracy of 0.05 mGals? Assume the voids are formed in
limestone (density 2.7 gm/cm3) and that void centers are
never closer to the surface than 100m.
Take a few minutes to work through this one.
Tom Wilson, Department of Geology and Geography
Consider another in-class problem
12 sectors with Ri=1100 and Ro=2200
Ring
The butte fits
into one sector
Tom Wilson, Department of Geology and Geography
Butte
Graphical separation of residual
Examine the map at right.
Note the regional and residual
(or local) variations in the
gravity field through the area.
The graphical separation
method involves drawing
lines through the data that
follow the regional trend.
The green lines at right extend
through the residual feature
and reveal what would be the
gradual drop in the anomaly
across the area if the local
feature were not present.
Mark off anomaly intersections with an
interpolated regional field
The residual anomaly is
identified by marking the
intersections of the extended
regional field with the actual
anomaly and labeling them with
the value of the actual anomaly
relative to the extended regional
field.
0
-1
After labeling all intersections
with the relative (or residual )
values, you can contour these
values to obtain a map of the
residual feature.
-0.5
-0.5
Circular in
shape … What
is the depth?
Just for general discussion > (see 6.8, Burger et al.): The curve in the
following diagram represents a traverse across the center of a roughly
equidimensional ore body. The anomaly due to the ore body is obscured by a
strong regional anomaly. Remove the regional anomaly and then evaluate the
anomaly due to the ore body (i.e. estimate it’s deptj and approximate radius)
given that the object has a relative density contrast of 0.75g/cm3
Bouguer Anomaly (mGal)
P roblem 5
0.00
-0.25
-0.50
-0.75
-1.00
-1.25
-1.50
0.0
0.5
1.0
1.5
Horizontal Position (km)
Tom Wilson, Department of Geology and Geography
2.0
You could plot the data on a sheet of graph paper. Draw a line
through the end points (regional trend) and measure the difference
between the actual observation and the regional (the residual).
You could use EXCEL or PSIPlot to fit a line to the two end points
and compute the difference between the fitted line (regional) and
the observations.
residual
Tom Wilson, Department of Geology and Geography
Just as with the graphical approach, the idea is to
remove the regional so you can investigate the residual.
Tom Wilson, Department of Geology and Geography
Are alternative
acceptable solutions
possible?
Tom Wilson, Department of Geology and Geography
Tom Wilson, Department of Geology and Geography
The large scale geometry of these density contrasts does not
vary significantly with the introduction of additional faults
Tom Wilson, Department of Geology and Geography
The differences in calculated gravity are too
small to distinguish between these two models
Tom Wilson, Department of Geology and Geography
Gravity applications span a variety of scales
Shallow
environmental
applications
Roberts, 1990
Tom Wilson, Department of Geology and Geography
http://pubs.usgs.gov/imap/i-2364-h/right.pdf
Tom Wilson, Department of Geology and Geography
The influence of near surface (upper 4 miles) does not explain
the variations in gravitational field observed across WV
c
c’
The paleozoic
sedimentary
cover
Morgan 1996
Tom Wilson, Department of Geology and Geography
The sedimentary cover plus variations in crustal thickness explain
the major features we see in the terrain corrected Bouguer
anomaly across WV
Morgan 1996
Tom Wilson, Department of Geology and Geography
In this model we incorporate a crust consisting of two layers: a
largely granitic upper crust and a heavier more basaltic crust
overlying the mantle
Morgan 1996
Tom Wilson, Department of Geology and Geography
Gravity model studies help us estimate the possible
configuration of the continental crust across the region
Derived from Gravity Model Studies
Tom Wilson, Department of Geology and Geography
Locate your swimming pool …
Ghatge, 1993
Tom Wilson, Department of Geology and Geography
Items on the list ….
•
•
•
•
•
•
Gravity paper summary(s) (both sections) due this Thursday,
Nov. 14th
Gravity lab will be due on Thursday November 21st (writing
section submission is self-reviewed showing track changes).
Keep working up the gravity lab
We’ll be getting into magnetics this Thursday so start
reading chapter 7.
There will be a short magnetics lab component. The lab
report will be less extensive than prior reports.
We will have two final exam review sessions: December 5th
and December 10th.
Tom Wilson, Department of Geology and Geography
Regular section submissions
All those in the regular section submit
paper copies of your paper summaries
and lab reports.
Tom Wilson, Department of Geology and Geography
Writing Section reminders
(electronic submissions only)
•
•
Revised paper summary 1 and self-reviewed paper summary
2 are to be turned in this Thursday.
The gravity lab is self reviewed and is due on Thursday,
November 21st.
All those in the writing section submit their
papers and lab electronically. Don’t forget to
turn on track changes while doing your selfreview. Only submit the self-reviewed file.
Tom Wilson, Department of Geology and Geography
What’s coming up?
Some due date reminders
Tom Wilson, Department of Geology and Geography