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)

Expanded Expanded VersionVersion

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

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.

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

2/1max

2

2

2

max

R

Zg

ZgR

Z

R

Z

R

Z

RGg

Horizontal Cylinder

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

0.58

0.71

0.710.58

11.421.74

Below are two symmetrical gravity anomalies. Which anomaly is associated with a buried sphere and which with the horizontal cylinder?

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 generate 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 0.14kilofeet. The range is 0.37kilofeet.

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

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.

(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

The half-plate or vertical fault.

We’ve talked a lot about edge effects in reference to Stewart’s paper and his use of the Bouguer plate term to estimate valley depth from the residual Bouguer anomaly.

We’ve examined those effects using model studies, but it turns out that the variation in acceleration across the edge of the plate has a fairly simple analytical expression.

Edge phenomena are common in exploration applications as fault related density contrasts in addition to those we encounter in Stewart’s paper (buried valley edges) that have application to ground water exploration.

We obtained a simple intermediate formula during our derivation of the acceleration associated with the infinite plate.

That formula -

2

22

ddzGg plate

when integrated over the limits -/2 to /2 represents the influence of a plate whose edges lie at infinity. It would be a simple matter to compute the effects of a more limited plate or sheet - one with a nearby edge, for example, or a faulted layer.

Diagnostic Position Depth Index Multiplier9/10 max 1/3 = 0.3333/4 max 1/1 = 11/2 max Fault edge1/4 max 1/1 = 1

1/10 max 1/3 = 0.333

Half Plate

(feet) 01277.0

(feet) 01277.0

(meters) 04190

(feet) 012770

max

max

max

max

t

g

gt

t .g

t.g

Can you determine what Z and t are using the variations in gravitational acceleration observed across the edge of the plate?

Comment on edge effects: The foregoing discussion of the variations in acceleration across the edge of a semi-infinite plate relate directly to the potential pitfalls associated with the use of the infinite plate formula 2Gt to estimate t - the depth of the buried valley.

First - note that the formula returns t and not z-t/2 or z+t/2 which would be the depth to the valley floor on the high and low sides of the scarp.

Secondly, sharp edges are transformed into rounded “shoulders” associated with the transition in the acceleration associated with an infinite plate to that where the plate is absent. This effect is certainly significant over the range 6Z.

More on edge effects - model illustrations

There are two additional simple geometrical objects that we mention only briefly -

•vertical cylinder

•vertical sheet

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 radiius 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

These simple geometrical objects allow you to interpret gravity anomalies in the cases where geometrical simplification is possible.

They also allow you to make estimates of what the shape and magnitude of a certain density configuration will be. Thus you might be able to determine or spec out the survey parameters you will need in order to detect a certain objective interval or target of certain geometry and density contrast - without having to run back to your computer.

Also recall the superposition principle from our earlier discussions. The superposition principle states that the resultant field arising from a combination of masses is simply the sum of the contributions of each mass taken separately. Thus -

i i

i

r

GMg

2

This idea applies not only to small points or differential elements comprising an object but to the objects themselves - thus -

-if we wished to determine what the combined effect would be of a spherical region of density contrast sitting above or below a density edge, we could compute them separately and then add them together to get the resultant field associated with that combination of objects.

Yuhr, 199

Yuhr, 199

Ghatge, 199

Ghatge, 199

Ghatge, 199

Ghatge, 199

Roberts, 199

Roberts, 199

Roberts, 199

Roberts, 199

Morgan 1996

Gurshaw, 1996

Morgan 1996

Morgan 1996

Morgan 1996