Chapter 3 Extracting Information from Geological Maps & Folds Geologic Maps Geologic maps are one of the most fundamental types of geologic docu- ments and yet they are a strange mixture of data and interpretation; it is often not easy to tell one from the other. Published maps are all solid colors and bold, confi- dent lines; they look like they are representing data. In this Chapter, we will build on the visualization that we began in Chapter 1 and the vector methods from Chapter 2 to introduce you to some of the types of information that you can ex- tract from a geologic map. Most of the features that we deal with at a map scale are approximately pla- nar features — stratigraphic contacts, faults, dikes, etc. — and, more often than not, they are not horizontal. At a regional scale, the surface of the earth where we make most of our observations is pretty planar but, at more detailed scales, topog- raphy is very irregular. Mapping the outcrop patterns of these planar features is ba- sically an exercise in locating the line of intersection between the planar feature and the irregular 3D surface of the Earth. CHAPTER 3 GEOLOGICAL MAPS Three Point Problems Some Common Map Symbols Contact A geologist cannot always go the the Fault field and put a compass on the rock to de- Thrust fault, saw teeth on upper plate termine a strike and dip. Even when s/he Fault, bar and ball on downthrown side can do so, we frequently want to know the orientation of a planar rock unit at a scale Anticlinal trace of axial surface larger than the outcrop scale. That is, we synclinal trace of axial surface want to know the average orientation at a Overturned anticline trace of axial map scale. Fortunately, our geologic maps surface with trend and plunge of hinge provide sufficient information to enable the Overturned syncline trace of axial surface Strike and Dip of Bedding, of overturned structural geologist to determine orientation bedding independent of outcrop and compass. Strike and Dip of Cleavage or foliation, of joint The orientation of a plane can be de- termined if we know the positions of three non-collinear points within the plane. Because the surface of the earth has variable topography, we can commonly find three points on a plane at different topographic elevations. Geologic maps com- 7600 ft slope distance Js −1 ⎛ 800 ft ⎞ δ = tan = 38° ⎝⎜ 1025.5 ft⎠⎟ 6800 ft map distance = 1025.5 ft 200 0! 1000 m Figure 3.1 — The classical three-point problem where a line is draw between two points of equal elevation on a planar surface, yielding the strike (left). On the right is the construc- tion in a vertical plane parallel to the line labeled “1025.5” (feet) on the map for calculating the dip using the elevation values (in feet) and map distance. MODERN STRUCTURAL PRACTICE "50 R. W. ALLMENDINGER © 2015-16 CHAPTER 3 GEOLOGICAL MAPS monly depict topography with contour lines and, even if they don’t, modern digital elevation models are available for most of the earth’s surface and the elevation of any point can be determined via the Internet. Figure 3.1 shows the classical way of determining strike and dip from a geo- logic map with topographic contours. This simple method takes advantage of the fact that a line connecting two points of equal elevation along a mapped contact of a planar feature define the strike of a plane. The dip can then be calculated from a third point at different elevation from a simple geometric construction. Both the map distance, and the elevation difference to the third point, perpendicular to the strike, can be read directly off the geologic map. A more general method takes advantage of the vector methods we’ve just learned in the previous chapter. It is more flexible because all three points can be at different elevations and can be used wherever we have spot elevations but no topo- graphic contours (e.g., in Google Earth). We use the cross product of two vectors in a plane — just like we did in the previous Chapter to determine the true dip from two apparent dips — but this time we are not using unit vectors describing orienta- tions of two lines in the plane, but are using position vectors whose magnitude is much greater than one (Fig. 3.2). A position vector is a line connecting a point in space to the origin of the co- ordinate system (P1, P2, and P3 in Fig. 3.2). The coordinates of the position vector are just the scalar components of the vector projected onto the coordinate system axes. In the case of our geologic map, we could use the UTM coordinates (eastings U Figure 3.2 — Three points in a plane can be N v used to calculate the orientation of a plane. P2 The coordinates of the three points are the u coordinates of the three position vectors, P1, P2, and P3. To get the vectors that lie P1 P3 within the plane, v and u, we use vector subtraction as described in the text. The pole to the plane is calculated from v × u. E MODERN STRUCTURAL PRACTICE "51 R. W. ALLMENDINGER © 2015-16 CHAPTER 3 GEOLOGICAL MAPS and northings) plus elevation to define the position vector in an East-North-Up co- ordinate system. Or, we could use any other local Cartesian coordinate system. To calculate the pole to the plane, we will use the cross product of two vec- tors in the plane, v and u. These two vectors can be calculated from the position vectors using vector subtraction. The complete sequence of steps is given, below: The first step is to subtract the position vectors to get v and u. Note that at this point, we are working in an ENU coordinate system so the subscripts in the fol- lowing equation correspond to the axes of our coordinate system: 1=E, 2=N, and 3=U. v ⎡ P2 P1 P2 P1 P2 P1 ⎤ = ⎢ ( 1 − 1 ) ( 2 − 2 ) ( 3 − 3 ) ⎥ ⎣ ⎦ (3.1) ⎡ ⎤ u = (P31 − P11 ) (P32 − P12 ) (P33 − P13 ) ⎣⎢ ⎦⎥ To convert these to a lower hemisphere NED centric coordinate system — which we will need in order to calculate our orientations — we switch the order of the first two components of the vector and multiply the third by –1: v ⎡ P2 P1 P2 P1 P2 P1 ⎤ ′ = ⎢ ( 2 − 2 ) ( 1 − 1 ) −( 3 − 3 ) ⎥ ⎣ ⎦ (3.2) ⎡ ⎤ u′ = (P32 − P12 ) (P31 − P11 ) −(P33 − P13 ) ⎣⎢ ⎦⎥ The cross product is defined as: " v′ × u′ = ⎡ v u − v u v u − v u v u − v u ⎤ = ⎡ s s s ⎤ (3.3) ⎣⎢ ( 2 3 3 2 ) ( 3 1 1 3 ) ( 1 2 2 1 ) ⎦⎥ ⎣ 1 2 3 ⎦ The cross product gives us the pole to the plane but we need to convert it to a unit vector before it can be transformed back into geographic orientations like trend and plunge or strike and dip. We start by calculating the magnitude of cross prod- uct, smagn: 2 2 2 " smagn = s1 + s2 + s3 (3.4) MODERN STRUCTURAL PRACTICE "52 R. W. ALLMENDINGER © 2015-16 CHAPTER 3 GEOLOGICAL MAPS And now we calculate the unit pole vector, " pˆ , by dividing each component of s by its magnitude, smagn. ⎡ s s s ⎤ " pˆ = ⎡ p p p ⎤ = ⎢ 1 2 3 ⎥ (3.5) ⎣ 1 2 3 ⎦ s s s ⎣⎢ magn magn magn ⎦⎥ If the third component is negative, i.e., p3 < 0, then the unit pole vector we have calculated points into the upper hemisphere. To covert to the lower hemisphere in this case, multiply each component by –1. We’re now ready to convert our unit pole vector back to trend and plunge. The plunge is straightforward because it is just the arcsine of the p3 component of the pole to the plane: −1 " plunge = sin ( p3 ) (3.6) As we saw in Chapter 2, the trend is a function of p2 and p1 and the sign of p1. If p1 ≥ 0 then you use the equation on the left, below; otherwise use the equation on the right: −1 ⎛ p2 ⎞ −1 ⎛ p2 ⎞ " trend = tan ⎜ ⎟ or trend = 180 + tan ⎜ ⎟ (3.7) ⎝ p1 ⎠ ⎝ p1 ⎠ To get the right-hand rule strike, just add 90° to the trend; the dip is 90°–plunge. Figure 3.3 — A three point calcu- lation in GMDE. This program can also calculate the uncertainties; those shown here are given a hori- zontal and vertical uncertainty of 40 ft. MODERN STRUCTURAL PRACTICE "53 R. W. ALLMENDINGER © 2015-16 CHAPTER 3 GEOLOGICAL MAPS Figure 3.4 — The spreadsheet to do the same calculation as shown in Figure 3.3. Note that the switch from an ENU to an NED coordi- nate system occurs in row 11. The program GeolMapDataExtractor (GMDE) can do this calculation for you automatically (Fig. 3.3), and the equations we have just seen are exactly how the program does it. As a scientist, you are, of course, not content to trust your pre- cious data to a canned program, and thus will want to calculate these values your- self. Figure 3.4 shows you how to set up your spreadsheet using the same values as in Figure 3.3. Stratigraphic Thickness from Maps Stratigraphic thickness is defined as the thickness of a unit measured per- pendicular to the upper and lower surfaces of the unit. For people truly interested in stratigraphic sequences, there is still no substitute for going out in the field and measuring a stratigraphic section with tape, compass, Jacob staff, or whatever. Mea- suring section that way is certainly the most accurate way to determine the thick- ness of a unit but the process is time-consuming.