Nettleton, 1971

Note that a particular anomaly, such as that shown below, could be attributed to a variety of different density distributions.

Note also, however, that there is a certain maximum depth beneath which this anomaly cannot have its origins.

gravity anomaly

What would be the maximum value of the residual anomaly over this model?Convert t=500 meters to feet.500 meters = 1640 feet.

g = -1640/130 milligals or

-12.6 milligals

12 tan2

xg G t

z

If we are only 700 meters from the edge - what would the computed depth be using the plate approximation?g = 9.5 milliGals.t = 130 g = 1245 feet

or 380 meters.

Remember -

1) the underlying assumption is that the drift valleys are much wider than they are deep,

2) that the expression t = 130g specifically uses the residual gravity to estimate drift thickness,

3) and in order to do that effectively, the reference value will have to be chosen carefully.

If bedrock rises to the surface at some point, that might be a good point to assign a value of 0 to the residual (depends on how extensive the outcrop is).

We can often estimate the gravitational acceleration associated with 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.

Estimates of maximum depth, density contrast, fault offset, etc. can often be made quickly and without the aid of a computer using simple relationships derived for simple geometrical forms.

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

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.

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 Rg

Z

R

Z

R

Z

g ZR

g Z

R

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.

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.

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

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 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.

Just as in the case of the anomaly associated with spherically distributed regions of subsurface density contrast, 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.

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

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

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

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 a given anomaly cannot have its origins.

Problem-

Determine which anomaly is produced by a sphere and which is produced by a 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

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

Nov. 10th Problems 1 & 2. Gravity lab

Nov. 15th Problems 4, 5, & 6.

Nov. 17thGravity paper summaries