Environmental and Exploration Geophysics II

Tom Wilson, Department of Geology and Geography

Environmental and Exploration Geophysics II

[email protected]

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Gravity Methods (VI) Gravity Methods (VI) wrap upwrap up


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-1500 -1000 -500 0 500 1000 1500

Distance from peak (m)

Bo

ug

uer

An

om

aly

(mG

als)

Sphere with radius R and density

X

z r


Sphere with radius R and density

X

z r

3

322 2

2

(4 / 3 ) 1

1

vert

G Rg

Z xz

gvert


3/ 22max

2

1

1

vgg x

z

This term defines the shape of the

anomaly produced by any spherically

shaped object.

Where 3

max 2

(4 / 3 )G Rg

Z

In this form, the relationship is normalized by the maximum value of g

observed at a point directly over the center of the sphere.


3/ 22max

2

1 1

21

vg

g xz

Solve for x1/2/z

We found that x1/2/z = 0.766

x½ is referred to as the diagnostic position, 1/x1/2 is referred to as the depth index multiplier

The Diagnostic Position


max

vg

g

3/4

1/2

1/4

That ratio can be solved for at any point on the curve to provide a diagnostic position.


A table of diagnostic positions and depth index multipliers for the Sphere (see your

handout).

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.

(depth index multiplier) times at the diagnostic positionZ X


3

max 2

3

2

3

2

1/32max

2max

3

(4 / 3 )

0.02793 for meters

0.00852 for feet

(feet)0.00852

(feet)0.00852

G Rg

Z

R

Z

R

Z

g ZR

g Z

R

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.

Once you figure out Z - Solve for R or 2

max3

(feet)(4 / 3 )

g Z

GR

1/32max (feet)

(4 / 3 )

g ZR

G


What is Z if you are given X1/3?

… Z = 0.96X1/3

In practice we can get as many estimates of Z as we have diagnostic positions. This allows you to estimate Z as a statistical average of several values.

Using the above table, we could make 5 separate estimates of Z. This allows the interpreter to evaluate how closely the object approximates the shape of a sphere.


You could measure of 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.

0.460.560.771.041.24

2.171.791.310.960.81

3/ 22

2

1

1xz


Using the depth index multiplier of 1.305, Z has to be 13.05 km

Since X3/4=z/2.17, then X3/4 =6km

Look at your

table for the

sphere


Based on the x1/2 distance and depth index multiplier of 1.305 what is z?

Based on the value of x3/4 and the depth index

multiplier of 2.17, What is z?

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-1500 -1000 -500 0 500 1000 1500


Bo

ug

uer

An

om

aly

(mG

als)

g1/2=0.175g3/4=0.26

gmax=0.35

1000 m

~2600m

~2700m


The Horizontal Cylinder

Refer to text for some background …


2

2

2

2 1

1cyl

G Rg

xZz

2

max

2 G Rg

Z

max 2

2

1

1cylg g

xz

Results for Horizontal Cylinder

and


We can ask the same kinds of questions we asked regarding the sphere. For example,

max

1

2cylg

gWhere does

2

2

1 1

2 1xz

2

2 1 2xz

2

2 1xz

1x

z

12

x z

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


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.

Horizontal Cylinder


How would you determine the depth index multipliers from this graph?

10.7

1.41

0.57

1.72


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 at the diagnostic position.

X times the depth index multiplier yields Z

X3/4X2/3

X1/2

X1/3X1/4

Z=X1/2

0.7

0.7

0.57

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

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.

Simple relationships and formula for the horizontal

cylinder


Nettleton, 1971

That maximum depth is a depth beneath which an anomaly of given wavelength cannot have a physical origin.

Maximum Depth

Non-uniqueness and the maximum depth


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

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?


(kilofeet) 52.8

(kilofeet) 52.8

3

2max

3/12max

R

Zg

ZgR

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.


Vertical Cylinder

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

2

1

2

1

1/ 2

max 1

max 12

0.01886 for meters

0.000575 for feet

(feet)0.000575

(feet)0.000575

R

Z

R

Z

g ZR

g Z

R


What simple geometrical object could be used to make a rough evaluation of these anomalies?

Sphere or vertical cylinder

Horizo

nta

l cy

lind

er


Half plate or faulted plate

How about the anomaly below?


Are alternative acceptable

solutions possible?


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

Shallow environmental applications


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


Morgan 1996

The influence of near surface (upper 4 miles) does not explain the variations in gravitational field observed across

WV

The paleozoic

sedimentary cover

c c’


Morgan 1996

The sedimentary cover plus variations in crustal thickness explain the major features we see in the terrain corrected

Bouguer anomaly across WV


Morgan 1996

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


Derived from Gravity Model Studies

Gravity model studies help us estimate the possible configuration of the continental crust across the region


Ghatge, 1993

It could even help you find your swimming pool


What is the radius of the smallest equidimensional void (such as a chamber in a cave - think of it more simply as an isolated spherical void) that can be detected by a gravity survey for which the Bouguer gravity values have an accuracy of 0.05 mG? Assume the voids are in limestone and are air-filled (i.e. density contrast, , = 2.7gm/cm3) and that the void centers are never closer to the surface than 100m.

i.e. z ≥ 100m

From last time


(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

Recall the formula developed for the sphere.

meters 02793.0

metersfor 02793.0

3/12max

2

3

ZgR

Z

R

Let gmax = 0.1

We reasoned that ganom

shouldbe at least 0.1 mGal; that Z would be at least 100m, and = 2.7 2.7gm/cm3 or 1.7gm/cm3


In a problem similar to problem 6.9 (Burger et al.) you’re 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 discussed in class today. 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


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)

In this in-class/take home problem determine whether the anomaly below is produced by a sphere of a cylinder


• Remember that paper summaries and gravity labs are due Thursday, November 15th

• Problems 6.5 and 6.9 are due Tuesday Nov. 13th.

• Begin reading Chapter 7 on Exploration using Magnetic Methods …

• We will introduce magnetic methods during the week of November 13th.

• No class next week

Environmental and Exploration Geophysics II

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Transcript of Environmental and Exploration Geophysics II