1. Gravity Methods (IV) Environmental and Exploration Geophysics II
tom.h.wilson
Department of Geology and Geography
West Virginia University
Morgantown, WV

2. Terrain Correction Problem
Reviewed

3. 200 520 2840 520 280 What’s the station elevation?
What’s the average elevation in Sector 1?
What’s the relative difference between the station elevation and the average elevation of sector 1?
200
520
2840
520
280

4. 2640
200
3 (0.03mG)
0.028mG
What did you get?
Determine the average elevation, relative elevation and T for all 8 sectors in the ring. Add these contributions to determine the total contribution of the F-ring to the terrain correction at this location.
We will also consider the F-ring contribution if the replacement density of 2.67 gm/cm3 is used instead of 2 gm/cm3 and the result obtained using equation 6-30

5. Equation 6-30

6. If you tried to use formula 6-30 to calculate the sector effects explicitly - did you remember to convert from feet to meters?
Remember that these corrections were made assuming a replacement density of 2 gm/cm3. If you wish to estimate the effect of topography assuming another replacement density then you must adjust the total sum by a factor equal to the ratio of the desired density contrast to 2.0 gm/cm3. Hence, in the present example, use of a 2.67 gm/cm3 replacement density requires that the sum ( milligals) be factored by the ratio 2.67/2.00 = Thus the total f-ring adjustment for 2.67gm/cm3 replacement density is 0.81 milligals.
You don’t have to hand this in but make sure you understand how to do it. Aspects of this procedure could appear on the final exam.

7. Graphical Separation of Residual
Examine the map at right. Note the regional and residual (or local) variations in the gravity field through the area.
The graphical separation method involves drawing lines through the data that follow the regional trend.
The green lines at right extend through the residual feature and reveal what would be the gradual drop in the anomaly across the area if the local feature were not present.

8. Turn in next time for extra credit
The residual anomaly is identified by marking the intersections of the extended regional field with the actual anomaly and labeling them with the value of the actual anomaly relative to the extended regional field. -1
-0.5
-0.5
After labeling all intersections with the relative (or residual ) values, you can contour these values to obtain a map of the residual feature.
Turn in next time for extra credit

9. Zero
Max = 0
Min ~-1.8
negative

10. Stewart makes his estimates of valley depth from the residuals
Stewart makes his estimates of valley depth from the residuals. You shouldn’t be concerned too much if you don’t understand the details of the method he used to separate out the residual, however, you should appreciate in a general way, what has been achieved. There are larger scale structural features that lie beneath the drift valleys and variations of density within these deeper intervals superimpose long wavelength trends on the gravity variations across the area. These trends are not associated with the drift layer.
The potential influence of these deeper layers is hinted at in one of Stewart’s cross sections.

11. Bouguer anomaly
Regional anomaly -
Residual anomaly =

12. If one were to attempt to model the Bouguer anomaly without first separating out the residual, the interpreter would obtain results suggesting the existence of an extremely deep glacial valley that dropped off to great depths to the west. However, this drop in the Bouguer anomaly is associated with the deeper distribution of density contrasts.

13. More on “Stewart’s law” - Stewart assumes that the residual anomaly can be directly related to drift thickness using the Bouguer plate equation.
Stewart assumes a density contrast of -  = 0.6 gm/cm3 and combines the terms 1/2G to yield the constant 130 which incorporates density units of grams/cm3 and thickness or depth (t) in units of feet

14. Stewart obtains the general relationships -or
Where Z, is the drift thickness at station , and
R is the value of the gravity residual at station 

15. From Tuesday's Lab
What would be the maximum value of the residual anomaly over this model?
Convert t=500 meters to feet.
500 meters = 1640 feet.
9 = -1640/130 milligals or
-12.6 milligals

16. 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.
Note how the reference point becomes super critical here. In order to get total depth rather than depth relative to the top of the valley wall, all these anomaly values need to be shifted down 11.3 milligals

17. It is important to realize that
1) that there is the underlying assumption 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 would be a good point to assign a value of 0 to the residual.

18. Edge Effects Last Chance for Questions on -
Gravity Lab Due Wednesday, Nov. 6th by 5pm

19. Simple Geometrical Objects

20. Simple Geometrical Objects
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.

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

28. gmax

29. Divide through by gmax

35. Highlight the fact that X measures distance from the anomaly peak, and is NOT an absolute reference along the profile line.

36. The “diagnostic position” is a reference location
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.

37. 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.

38. 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 ) assume that depths and radii are in the specified units (feet or meters), and that density is always in gm/cm3.

39. 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.

40. 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.

47. Now compute variation of g across the cylinder
and consider only the vertical component.
take vertical component

52. 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.

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

54. 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
X1/2
X1/3
X1/4
Z=X1/2

55. Again, note that these constants (i. e
Again, note that these constants (i.e ) assume that depths and radii are in the specified units (feet or meters), and that density is always in gm/cm3.

56. 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.
Nettleton, 1971

57. In-class problem-
Determine which anomaly is produced by a sphere and which is produced by a cylinder.

58. 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?

59. Z is in kilofeet, and  is in gm/cm3.
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.

60. Vertical Cylinder ? A A'

61. Second Gravity Problem Set
We will spend more time on simple geometrical objects during the next lecture, but for now let’s spend a few moments and review the problems that were assigned last lecture.
Pb. 4 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.

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

63. Pb. 5: 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

64. You could plot the data on a sheet of graph paper
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.
residual
Regional

65. In problem 6 your given three anomalies
In problem 6 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 discussed in class today. Carefully consider where the anomaly drops to one-half of its maximum value. Assume a minimum value of 0. A. B. C.

66. Due Dates Nov. 11th Problems 1-3.
Interactive modeling exercises: up to 10 points extra credit
Residual separation exercise: up to 5 points extra credit
Nov. 16th
Terrain correction
Gravity lab
Nov. 18th
Problems 4, 5, & 6.
Gravity paper summaries