Topographic Map
Interpretation

TECHNIQUE NUMBER 6

SLOPE ANGLE

(X and/or Y), Z axes

Slope Angle. One of the most common needs in quantitative map interpretation is to determine the angle of a slope. Slope angle can be used in the understanding of process (such as streamflow) or hazard (such as slope failure) or structure (such as the dip of a rock unit).

The calculation of slope angle is the first technique which is clearly a combination of those techniques previously discussed, namely relief (Technique number 3) and horizontal distance (Technique number 4) determinations. The relationship between them is:

Relief / Horiz. Distance = tan(Slope Angle).

For example: the average slope angle between Valley Center School (SW¼ of the SW¼...sec 22, T1S, R5E) on the Bozeman quad and the Fowler School (Se¼, SE¼...sec 22, T2S, R5 E)  on the same quad is:

Note:

• The slope angle is only accurate if the two points used to define the slope are connected by a line which is perpendicular to the contours (that is, that one point lies directly downslope from the other).

Dip of a Rock Unit . A variant of the slope calculation is the calculation of the dip of a rock unit. Note that topographic slope and geologic dip are often similar, but are not necessarily the same. (Dip tends to be parallel to and steeper than slope, as we will see later.) A major difficulty in the calculation of dip is the accurate estimation of slope direction, as the dipping beds are often highly cut up by erosional processes. The method we use to minimize that problem is the calculation of a three-point problem.

Note:

• The orientation of a rock unit (locally a plane in space, even if regionally deformed by folding and/or faulting) can be uniquesly defined by two characteristics: strike and dip.   The strike is the orientation of the line marking the intersection of the plane of the rock unit with a horizontal plane.  Think of strike as a "bathtub ring" or as a contour line on the surface of the rock.  Strike is bidirectional (see Technique Number 5) and is by convention reported in the northern hemisphere (e.g., N33°W, or 325° azimuth).  Dip is perpendicular to strike (thus can have only two directions once strike is known) and is by convention reported by the cardinal compass direction and the angle (e.g., 23° to the E).

A three-point problem requires three points, of known location and elevation, which lie ON THE SAME PLANE. In a geological context, they will probably lie on the surface which was the contact between a weak unit (now stripped away) and a resistant unit. The accuracy of the method requires that great care be taken to ensure that they lie on the same plane. The emphasis on plane underlines the assumption that the surface is planar, that is, that it is not folded. This means that the technique should only be applied over short distances, unless the major geological structures are clearly planar.

To solve a three-point problem:

the two points with the highest and lowest elevations should be joined with a light, thin pencil line (Line 1). (If two of the points have the same elevation, this step and the following one are not necessary.) The elevation on that line which matches that of the third point must be interpolated (by calculating the relief between the two points, determining the elevation of the intermediate point as a percentage of that relief, and scaling that percentage of the distance onto the line) and marked on the line (4734). [On a rock unit, this is strike.] The line connecting the interpolated point with the third point (Line 2) is therefore horizontal, and the perpendicular line from Line 2 to either of the first two points (Line 3) defines the slope direction. The slope of Line 3, or dip angle of a rock unit, can be calculated from the relief between Line 2 and the selected point and the length of line 3. [Dip direction is downhill, by convention.]

Slope Direction. As should be obvious from the discussion above, slope direction is a byproduct of the calculation of slope angle. For land surface slopes, the direction can be calculated by simply drawing a perpendicular to the contour lines and determining its direction using Technique Number 5.

For example, on the map below (Indian Hills, CO; originally 1:24,000 but enlarged) the strike of the small hogback XYZ is defined by the edge of the hogback at the 6200' contour (points X and Y) is N27°W - assuming N-S section boundary) and the dip is to the E (towards point Z, also on the edge of the hogback - invtan(160'/220') = 36° - assuming sec 31/32 boundary is 1 mile long). If overlying rocks mask the top of the resistent unit, then the topographic slope may underestimate the geologic dip.

EXERCISE: On the map segment reproduced at right, determine the slope angle along line A-A' and the strike and dip of the rock unit which controls the topography.