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 Gravity Methods (V)(V)


The question of edge effects addressed above takes advantage of simple geometrical objects, the plate and half plate, to answer questions about possible anomaly magnitudes associated with the problem at hand.

See Stewart et al., section 6.5 pages 378 to 397


We make simplifying assumptions about the geometry of complex objects such as dikes, sills, faulted layers, mine shafts, cavities, caves, culminations and anticline/syncline structures by approximating their shape using simple geometrical objects - such as horizontal and vertical cylinders, the infinite sheet, the sphere, etc. to estimate the scale of an anomaly we might be looking for or to estimate maximum depth, density contrast, fault offset, etc. without the aid of a computer.

Berger gets into a lot of the details in Section 6.5 of the text


Let’s start with one of the simplest of geometrical objects - the sphere -

Go to 31





What is the vertical component?

cossphV gg

1/ 22 2cos z

x z


3

32 2 2

43v

zg GRx z

3

32 2 2

43

v

GR zg

x z

33

2 23

2

43

1v

zg GRxzz

Hence ….


3

322 2

2

413

1v

GRg

z xz

3

2

43

v

GRg

z

zg directly over the

center of the sphere is gmax


gmax

32 2

2

1

1vg K

xz

3

max2

43

GRK g

z


Divide through by gmax

32 2max

2

1

1

vgg x

z

gmax contains information about volume, density and

radius


The shape of the curve gv/gmax is scale independent. It is not affected by the depth or

size of the sphere.


Shape of the anomaly is independent of the size of the sphere that produced it.The shape, the variation as a function

of x/z is the same for all spheres regardless of their depth or size.


At what point does the anomaly fall off to one-half of its maximum

value?


Let the ratio g/gmax = ½ and solve for X/Z


X1/2 /Z = 0.766 implies that Z can be expressed in terms of

X1/2


X measures distance from the anomaly peak, and is NOT an absolute reference along the

profile line. The location of the peak corresponds to X=0


The “diagnostic position” is a reference location. It refers to the X location of points where the anomaly has fallen to a certain fraction of its maximum value, for example, 3/4 or 1/2.


In the above, the “diagnostic position” is X1/2, or the X location where the anomaly falls to 1/2 of its

maximum value. The value 1.31 is referred to as the “depth index multiplier.” This is the value that you multiply the reference distance X1/2 by to obtain an

estimate of the depth Z.


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

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 RgZ

RZRZ

g ZR

g ZR

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 – to Z.

Note that these constants (e.g. 0.02793) assume that depths and radii are in the specified units (feet or meters), and that density is always in gm/cm3.


What is Z if you are given

X1/3?… Z = 0.96X1/3

In general you will get as many estimates of Z as you have diagnostic positions. This allows you to estimate Z as a statistical average of several values.We can make 5 separate estimates of Z given the diagnostic position in the above table.


You could measure off 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.

1.3050.96

1.79

0.81

2.17See your handout


Given X1/2 what is Z?


The Cylinder

Refer to text for details …


Some details on the following slides …


2

22

2 1

1cyl

G RgxZ

z

2

max2 G Rg

Z

max 22

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

12

cylgg

Where does

22

1 12 1x

z

22 1 2x

z

22 1x

z

1xz

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. X times the depth index multiplier yields Z

X3/4 X2/3

X1/2X1/3X1/4

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

RZg

ZgR

ZR

ZR

ZRGg

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.


Nettleton, 1971

We should note again, that the depths we derive assuming these simple geometrical objects are maximum depths to the centers of these objects - cylinder or sphere. Other configurations of density could produce such anomalies.

This is the essence of the limitation we refer to as non-uniqueness. Our assumptions about the actual configuration of the object producing the anomaly are only as good as our geology.

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

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

RZg

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.


We will spend more time on simple geometrical objects during the next lecture, but for now let’s spend a few moments and review some additional problems from

the text Pb-1: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 void centers are never closer to the surface than 100.


(feet) 00852.0

(feet) 00852.0

feetfor 00852.0

metersfor 02793.0

)3/4(

3

2max

3/12max

2

3

2

3

2

3max

RZg

ZgR

ZR

ZR

ZRGg

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

meters02793.0

metersfor 02793.0

3/12max

2

3

ZgR

ZR


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

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 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

Distance from peak (m)

Boug

uer A

nom

aly

(mG

als)

A.

C.B.


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

• Problems 6.5 and 6.9 are due Tuesday, November 13th.

• Begin reading Chapter 7 on Exploration using Magnetic Methods …

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

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Environmental and Exploration Geophysics II