Gravity Methods (VII) more wrap up

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

2 Bouguer Anomaly (mgals) Distance from peak (m) X z r Sphere with radius R and density

3 g vert 3 G(4/3 R ) 1 3 Z X x z 1 g vert z r Sphere with radius R and density

4 Diagnostic position X 1/ g g v max 1 1 x1/ 1 z 3/ x ½ is referred to as the diagnostic position, 1/x 1/ is referred to as the depth index multiplier We solve for x 1/ /z and find that x 1/ /z =

5 The values at several points along the X/Z axis can be calculated to define the shape of the anomaly g g v max 3/4 1/ 1/4

6 The other day we developed these tables of diagnostic positions and depth index multipliers Diagnostic Position Depth Index Multiplier (g/gmax) 3/4 max 1/0.46 =.17 /3 max 1/0.56 = / max 1/0.77 = /3 max 1/1.04 = /4 max 1/1.4 = 0.81 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. Z (depth index multiplier) times X at the diagnostic position

7 Check it out. Depth index multiplier for X 1/ is Depth index multiplier for X 3/4 is.17 What depth do you get? X 1/ X 3/4 m Wilson, Department of Geology and Geography

8 Once you figure out Z - Solve for R or 3 G(4/3 R ) gmax Z 3 R for meters Z 3 R for feet Z gmaxz R gmaxz R 3 1/3 1/3 gmax Z R (feet) (4/3 ) G (feet) (feet) gmaxz (feet) 3 (4/3 ) GR These constants (i.e or ) assume that depths and radii are in the specified units (feet or meters), and that density is always in gm/cm 3.

9 X z Cylinder with radius R and density R

10 X z r At surface distance x away from a point directly over the cylinder

11 1 1 cyl G R g x Z z max G R g Z max 1 1 cyl g g x z Results for Horizontal Cylinder and

12 We can ask the same kinds of questions we asked regarding the sphere. For example, Where does g cyl g max 1 1 x z 1 1 x 1 z x z 1 x z 1 x1 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

13 X 3/4 X /3 X 1/ Locate the points along the X/Z Axis where the normalized curve falls to diagnostic values - 1/4, 1/, etc. The depth index multiplier is just the reciprocal of the value at X/Z at the diagnostic position. X times the depth index multiplier yields Z X 1/4 X 1/ Z=X 1/

14 Simple relationships and formula for the horizontal cylinder Diagnostic Position Depth Index Multiplier 3/4 max 1/0.58 = 1.7 /3 max 1/0.71 = / max 1/1= 1 1/3 max 1/1.4 = 0.7 1/4 max 1/1.74 = 0.57 GR gmax Z R for meters Z R for feet Z gmaxz R gmaxz R 1/ (feet) (feet) With Z, you can then speculate on the density contrast or radius of the object in question. 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.

15 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.

16 Assume the anomaly below is produced by long horizontal tunnel What is the depth to the tunnel? What are the depth index multipliers? X 1/ X 3/4

What is the approximate shape of the object producing the anomaly below? Diagnostic positions Multipliers Sphere Z Sphere Multipliers Cylinder X 3/4 = 0.95.17.06 1.7 1.63 X /3 = 1.15 1.79.06 1.41 1.

17 What is the approximate shape of the object producing the anomaly below? Diagnostic positions Multipliers Sphere Z Sphere Multipliers Cylinder X 3/4 = X /3 = X 1/ = X 1/3 = X 1/4 = Z Cylinder 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 Estimating other properties of the buried object 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/cm 3. R gmaxz 8.5 g max Z 8.5R 3 1/ 3 (kilofeet) (kilofeet)

19 For the remainder of the class consider the inclass activity sphere cylinder 10 10

Vertical Cylinder Diagnostic Position Depth Index Multiplier 3/4 max 1/0.86 = 1.16 /3 max 1/1.1 = 0.91 1/ max 1/1.7= 0.58 1/3 max 1/.76 = 0.

20 Vertical Cylinder Diagnostic Position Depth Index Multiplier 3/4 max 1/0.86 = 1.16 /3 max 1/1.1 = / max 1/1.7= /3 max 1/.76 = /4 max 1/3.7 = 0.7 g G R 1 1 z x z L x 1/ 1/ R for meters Z 1 R for feet Z gmaxz1 R gmaxz R 1 1/ (feet) (feet)

21 For a given anomaly certain simple geometries can be assumed and tested Horizontal cylinder or vertical dyke Sphere or vertical cylinder

22 How about the anomaly below? Half plate or faulted plate

23 Problem What is the radius of the smallest equidimensional void (e.g. chamber in a cave) that can be detected by a gravity survey for which the Bouguer gravity values have an accuracy of 0.05 mgals? Assume the voids are formed in limestone (density.7 gm/cm 3 ) and that void centers are never closer to the surface than 100m. Take a few minutes to work through this one.

24 Consider another in-class problem 1 sectors with R i =1100 and R o =00 Ring The butte fits into one sector Butte

25 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.

26 Mark off anomaly intersections with an interpolated regional field 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. After labeling all intersections with the relative (or residual ) values, you can contour these values to obtain a map of the residual feature

27 Circular in shape What is the depth?

28 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/cm 3 Bouguer Anomaly (mgal) Problem Horizontal Position (km)

29 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

30 Just as with the graphical approach, the idea is to remove the regional so you can investigate the residual.

31 Are alternative acceptable solutions possible?

33 The large scale geometry of these density contrasts does not vary significantly with the introduction of additional faults

34 The differences in calculated gravity are too small to distinguish between these two models

35 Gravity applications span a variety of scales Shallow environmental applications Roberts, 1990

37 The influence of near surface (upper 4 miles) does not explain the variations in gravitational field observed across WV c c The paleozoic sedimentary cover Morgan 1996

38 The sedimentary cover plus variations in crustal thickness explain the major features we see in the terrain corrected Bouguer anomaly across WV Morgan 1996

39 In this model we incorporate a crust consisting of two layers: a largely granitic upper crust and a heavier more basaltic crust overlying the mantle Morgan 1996

40 Gravity model studies help us estimate the possible configuration of the continental crust across the region Derived from Gravity Model Studies

41 Locate your swimming pool Ghatge, 1993

42 Items on the list. Gravity paper summary(s) (both sections) due this Thursday, Nov. 14th Gravity lab will be due on Thursday November 1 st (writing section submission is self-reviewed showing track changes). Keep working up the gravity lab We ll be getting into magnetics this Thursday so start reading chapter 7. There will be a short magnetics lab component. The lab report will be less extensive than prior reports. We will have two final exam review sessions: December 5 th and December 10 th.

43 Regular section submissions All those in the regular section submit paper copies of your paper summaries and lab reports.

44 Writing Section reminders (electronic submissions only) Revised paper summary 1 and self-reviewed paper summary are to be turned in this Thursday. The gravity lab is self reviewed and is due on Thursday, November 1 st. All those in the writing section submit their papers and lab electronically. Don t forget to turn on track changes while doing your selfreview. Only submit the self-reviewed file.

45 What s coming up? Some due date reminders

Intro to magnetic methods

Environmental and Exploration Geophysics I Intro to magnetic methods tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV Items on the list Gravity