Environmental and Exploration Geophysics II

t.h. wilsonwilson@geo.wvu.edu

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Gravity Methods Gravity Methods (V)(V)

Diagnostic Position Depth Index Multiplier3/4 max 1/0.46 = 2.172/3 max 1/0.56 = 1.791/2 max 1/0.77 = 1.3051/3 max 1/1.04 = 0.961/4 max 1/1.24 = 0.81

(feet) 00852.0

(feet) 00852.0

feetfor 00852.0

metersfor 02793.0

)3/4(

3

2max

3/12max

2

3

2

3

2

3

max

R

Zg

ZgR

Z

R

Z

R

Z

RGg

The Sphere

You could measure the values of the depth index multipliers yourself from this plot of the normalized curve that describes the shape of the gravity anomaly associated with a sphere.

2

3

max

)3/4(

Z

RGg

32 2max

2

1

1

gshape term

g xz

The Sphere

Diagnostic Position Depth Index Multiplier3/4 max 1/0.58 = 1.722/3 max 1/0.71 = 1.411/2 max 1/1= 11/3 max 1/1.42 = 0.71/4 max 1/1.74 = 0.57

(feet) 01277.0

(feet) 01277.0

feetfor 01277.0

metersfor 0419.0

2

2max

2/1

max

2

2

2

max

R

Zg

ZgR

Z

R

Z

R

Z

RGg

Horizontal Cylinder

2

2

1

1xz

X3/4X2/3

X1/2

X1/3X1/4

Z=X1/2

Locate the points along the X/z Axis where the normalized curve falls to diagnostic values - 1/4, 1/2, etc.

The depth index multiplier is just the reciprocal of the value at X/Z.

X times the depth index multiplier yields Z

1.72

1.41

0.710.58

10.70.57

Depth Index

Multipliers

Z

RGg

2

max

2

2max

2

1

1

gshape term

g x

z

We left you with questions about these two anomalies last Thursday. Which anomaly is associated with a buried sphere and which with the horizontal cylinder?

Sphere:__?_ or Cylinder _?__

What is the depth Z?

If = 0.1 what is R ____?

Sphere:__?_ or Cylinder _?__

What is the depth Z?

If = 0.1 what is R ____?

Diagnosticpositions

MultipliersSphere

ZSphere MultipliersCylinder

ZCylinder

X3/4 = 0.95 2.17 2.06 1.72 1.63X2/3 = 1.15 1.79 2.06 1.41 1.62X1/2 = 1.6 1.305 2.09 1 1.6X1/3 = 2.1 0.96 2.02 0.7 1.47X1/4 = 2.5 0.81 2.03 0.57 1.43

The standard deviation in the estimates of Z assuming that you have a sphere is 0.027kilofeet. The range is 0.06 kilofeet.

When you assume that the anomaly is generated by a cylinder, the range in the estimate is 0.2 kilofeet and the standard deviation is 0.093 kilofeet.

Assuming that the anomaly is generated by a sphere yields more consistent estimates of Z.

Diagnosticpositions

MultipliersSphere

ZSphere MultipliersCylinder

ZCylinder

X3/4 = 1.2 2.17 2.6 1.72 2.06X2/3 = 1.4 1.79 2.5 1.41 1.97X1/2 = 2 1.305 2.6 1 2X1/3 = 2.84 0.96 2.72 0.7 1.99X1/4 = 3.54 0.81 2.87 0.57 2.02

The standard deviation in the estimates of Z assuming that you have a sphere is 0.14 kilofeet. The range is 0.37kilofeet.

When you assume that the anomaly is generated by a cylinder, the range in the estimate is 0.09 kilofeet and the standard deviation is 0.03 kilofeet.

Assuming that the anomaly is generated by a cylinder, in this case, yields more consistent estimates of Z.

Diagnosticpositions

MultipliersSphere

ZSphere MultipliersCylinder

ZCylinder

X3/4 = 0.95 2.17 2.06 1.72 1.63X2/3 = 1.15 1.79 2.06 1.41 1.62X1/2 = 1.6 1.305 2.09 1 1.6X1/3 = 2.1 0.96 2.02 0.7 1.47X1/4 = 2.5 0.81 2.03 0.57 1.43

Diagnosticpositions

MultipliersSphere

ZSphere MultipliersCylinder

ZCylinder

X3/4 = 1.2 2.17 2.6 1.72 2.06X2/3 = 1.4 1.79 2.5 1.41 1.97X1/2 = 2 1.305 2.6 1 2X1/3 = 2.84 0.96 2.72 0.7 1.99X1/4 = 3.54 0.81 2.87 0.57 2.02

If we take the average value of Zsphere as our estimate we obtain Z=2.05kilofeet which we can round off to 2kilofeet

If we take the average value of Zcyl as our estimate we obtain Z=2 kilofeet.

We left you with questions about these two anomalies last Thursday. Which anomaly is associated with a buried sphere and which with the horizontal cylinder?

Sphere: _X_ or Cylinder __

Depth Z = 2kf

If = 0.1 what is R ____?

Sphere:___ or Cylinder _X__

Depth Z = 2kf

If = 0.1 what is R ____?

(kilofeet) 77.12

(kilofeet) 77.12

2max

2/1max

R

Zg

ZgR

(kilofeet) 52.8

(kilofeet) 52.8

3

2max

3/12max

R

Zg

ZgR

Sphere

Cylinder

For the sphere, we find that R = 1 kilofoot

For the cylinder, we find that R is also = 1 kilofoot

If = 0.1gm/cm3

Diagnostic Position Depth Index Multiplier3/4 max 1/0.86 = 1.162/3 max 1/1.1 = 0.911/2 max 1/1.72= 0.581/3 max 1/2.76= 0.361/4 max 1/3.72= 0.27

Ztop

Zbottom

2R

(feet) 000575.0

(feet) 000575.0

metersfor 01886.0

feet 000575.0

2max

2/1max

2

max

2

max

R

Zg

ZgR

Z

Rg

Z

Rg

top

top

top

top

Vertical Cylinder

Note that the table of relationships is valid when Zbottom is at least 10 times the depth to the top Ztop, and when the radius of the cylinder is less than 1/2 the depth to the top.

Diagnostic Position Depth Index Multiplier3/4 max 1/1.48 = 0.682/3 max 1/1.96 = 0.511/2 max 1/3.16 = 0.321/3 max 1/5.16 = 0.191/4 max 1/6.65 = 0.15

(feet) 00936.0

(meters) 0307.0

)3026.2(2

)1/10ln(2

)/ln(2

max

max

12max

Wg

Wg

WG

WG

ZZWGg

Z1

Z2W

The above relationships were computed for Z2=10Z1 and W is small with respect to Z1

Vertical Sheet

The large scale geometry of these density contrasts does not vary significantly with the introduction of additional faults

The differences in calculated gravity are too small to distinguish between these two models

Roberts, 1990

Estimate landfill thickness

http://pubs.usgs.gov/imap/i-2364-h/right.pdf

Morgan 1996

Morgan 1996

Morgan 1996

Derived from Gravity Model Studies

Ghatge, 1993

It could even help you find your swimming pool

3. 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. (Problem 6.5 from Burger et al.)

What simple geometrical object could be used to help you answer this questions?

What size anomaly are you trying to detect?

What equation should you use?

3

max 2

(4 / 3 )G Rg

Z

3. What is the radius …..?

(feet) 00852.0

(feet) 00852.0

feetfor 00852.0

metersfor 02793.0

)3/4(

3

2max

3/12max

2

3

2

3

2

3

max

R

Zg

ZgR

Z

R

Z

R

Z

RGg

Begin by recalling the list of formula we developed for the sphere.

(feet) 02793.0

metersfor 02793.0

3/12max

2

3

ZgR

Z

R

What are your givens?

& Z

Pb. 4: 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 depth and approximate radius) given that the object has a relative density contrast of 0.75g/cm3 with

surrounding strata.

Horizontal Position (km)

0.0 0.5 1.0 1.5 2.0

Bou

guer

Ano

mal

y (m

Gal

)

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

Problem 5

residual

Regional

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.

In problem 5 your given three anomalies. These anomalies are assumed to be associated with three buried spheres. Determine their depths using the diagnostic positions and depth index multipliers we’ve been discussing in class. Carefully consider where the anomaly drops to one-half of its maximum value. Assume a minimum value of 0.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-1500 -1000 -500 0 500 1000 1500

Distance from peak (m)

Bo

ug

uer

An

om

aly

(mG

als)

A.

C.

B.

Anomaly from object of unknown geometry

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1000 1500 2000 2500 3000

Distance (meters)

An

om

aly

(mG

als)

Given = 1 gm/cm3

• Problems 6.1 through 6.3 are due today Tuesday, Nov. 6th.

• Hand in gravity lab this Thursday, Nov. 8th .

• Turn in Part 1 (problems 1 & 2) of gravity problem set 3, Thursday, November 8th. Remember to show detailed computations for Sector 5 in the F-Ring for Pb. 2.

• Turn in Part 2 (problems 3-5) of gravity problem set 3, Tuesday, November 13th.

• Gravity paper summaries, Thursday, November 15th.