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Diagnostic Positions and Depth Index Multipliers for the Sphere

This document provides a table of diagnostic positions and depth index multipliers for the sphere, allowing for the estimation of depth and other parameters. It also discusses the shape of anomalies produced by spherical and cylindrical objects and how to determine depth index multipliers from a graph.

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Diagnostic Positions and Depth Index Multipliers for the Sphere

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  1. 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.

  2. More on Simple Geometrical Objects This term defines the shape of the anomaly produced by spherically shaped objects

  3. ½ is referred to as the diagnostic position, z/x is referred to as the depth index multiplier Solve for x/z We found that x/z = 0.766

  4. 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.

  5. 0.56 1.79 0.77 1.31 1.04 0.96 1.24 0.81 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. 0.46 2.17

  6. Based on the x1/2 depth index multiplier of 1.305 we get z = 13.05 km The depth index multiplier for x3/4 is 2.17/ What is z?

  7. Z has to be 13.05 km What is x3/4 = ? X3/4=z/2.17 =6km

  8. Let the diagnostic position (dp) be any other fraction (gv/gmax) and solve for additional diagnostic relationships No matter what diagnostic position you use, the z should be the same. If your assumption about the geometry of the object is correct.

  9. 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.

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

  11. 0.58 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 X2/3 X3/4 X1/2 X1/3 X1/4 Z=X1/2 1.72 1 1

  12. 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.

  13. 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

  14. Problem- Determine which anomaly is produced by a sphere and which is produced by a cylinder.

  15. Assume the anomaly is produced by a sphere. gmax=0.213 mGals ½gmax=0.107 What is x1/2?

  16. Assume the anomaly is produced by a sphere. gmax=0.213 mGals ½gmax=0.107 What is x1/2?

  17. 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?

  18. 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.

  19. Vertical Cylinder Ztop Zbottom 2R 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.

  20. Z1  W Z2  The above relationships were computed for Z2=10Z1 and W is small with respect to Z1

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

  22. Half Plate

  23. Second Gravity Problem Set 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 100m.

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

  25. 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

  26. 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.

  27. 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.

  28. Due Dates • Problems 6.1 through 6.3 are due today Tuesday, Nov. 6th. • Hand in gravity lab this Thursday, Nov. 8th . • Turn in Part 1 (problems 1 & 2) of gravity problem set 3, Thursday, November 8th. Remember to show detailed computations for Sector 5 in the F-Ring for Pb. 2. • Turn in Part 2 (problems 3-5) of gravity problem set 3, Tuesday, November 13th. • Gravity paper summaries, Thursday, November 15th.

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