• We often need to represent 3-d data on a 2-d sheet of paper. A contour map is a prime example.
• We usually have sparse, discrete data points; have to interpolate somehow in between.
  • Example: depth of a structure from drillholes.
  • Counter-example: deformation contours seen directly in interferometric radar maps.
  We must make assumptions of continuity. This is not always the case; faults can add discontinuities that are crucial to account for in a contour map. Only the most expensive contouring software can include fault discontinuities. You might do much better contouring such a map by hand.
  An irregular distribution of elevation control points.
• The quality of the map will be related to the density of control points.
• The extrapolation process requires the exercise of judgement and/or bias. A contour map is an interpretation, though often presented as data (even true of USGS topographic maps).
  The added knowledge of stream gullies substantially alters the interpreted contours, even though they are drawn from the same control points.
• Contouring is an application of interpolation.
• The distribution of control points can be:
  • Uniform - point density constant in any subarea of constant size.
  • Regular - points on a grid. Can be anisotropic, meaning more frequent in one direction than another.
  • Random - point locations have no relation to each other.
  • Clustered - can be in areas or in traverses.
• There are statistical measures of uniformity and the nearness of neighbors.
• Extrapolation of unequally-spaced control-point data to a grid, enabling contouring, can be accomplished with several different methods. The resources below detail extrapolation methods.
• Additional resources:
  1. Karlin's quick comparison of gridding methods, in HTML or PDF.
  2. Explanation of gridding methods from the Surfer manual, in HTML or PDF.

1. Spatial Aliasing Exercises: We're going to look at the differences between the two gravity anomaly maps below, which cover the same area and were contoured the same way, but from different data sets. Both have a 0.5 milligalileo (mGal) contour interval, although the color scheme differs between them. Small light-blue crosses mark the control points.
   1980s version of gravity map north of Reno, from USGS archives, above. Below is map of 1990s gravity data collected by Washoe County, with many more control points. Click on each map for a large JPEG image.
   1. Identify a well-sampled east-west profile about 5 km long where the two maps just don't match. It is common to find one generation of gravity map off by a constant shift from another generation of gravity map. Propose a value (in mGal) for the average offset between the two maps.
   2. Identify a feature detailed by the county map that was missed by the USGS map, because of a lack of 1980s control points. Along a 5 km west-east or south-north profile, plot two profiles on the same distance scale (adding your proposed constant offset between the maps, to one of the profiles). On the profiles plot the position and value of each contour line crossed, and mark the projected positions of only the nearest control points from the different maps. Use a different color for the features from each map. Make your profile with a ruler, to get distances of contour lines and control points along the profile, and a square, to project the nearby control points to the profile.
      Comparing the two profiles, what you are looking at is an example of spatial aliasing. Along your profile, the old USGS map didn't sample frequently enough to catch the low, or high, that is plain in the County map. You might notice that a feature will be missed unless it has been sampled by at least two points per wavelength. Another way of saying this is that a feature that is smaller than twice the local spacing will be invisible (and aliased out).

2. Hand Contouring Exercises: Click on each of the maps below for a printable PDF page.
   1. Hand-contour the topography in MAP1 below. Justify your selection of a contour interval. Be sure to label enough contours to make the map understandable, and properly hachure any closed depressions.
      (Larger GIF image)
   2. MAP2 gives the 2-way vertical reflection travel times to a structural horizon. Contour the map in time. Assuming constant velocity above the horizon, where are the synclines? If you are told that the area may contain NW-striking vertical normal faults, can you find one on this map? What is its throw in seconds? Conversely, if you know the structure is flat, where are the high-velocity zones? Again, be sure to label enough contours, and hachure any closed structural depressions.
      (Larger GIF image)

3. Computer Contouring Exercises: Download the text data file dixie-grav.txt and import it into the Surfer application in the MMV lab. Take care in importing the data; each line of the text file is in the order Latitude, Longitude, and Anomaly. This means that for a sensible map, it is in the order Y, X, and Anomaly. These coordinates plot the map below, leaving north toward the top.
   Map of gravity measurement locations in southern Dixie Valley, Nevada.
   1. Make a contour map, complete with all proper labels, and justify your choice of contour interval. Use Surfer's defaults.
   2. Change the gridding/interpolation method to an inverse-distance weighting technique (as described in the Surfer manual linked above). Describe the differences from the first map.
   3. Change the gridding/interpolation method back to kriging, but recognize the anisotropy of the geologic setting of these data. They were taken, as you can see from the map, along dip profiles crossing a north-striking basin. Use the suggestions in the Surfer manual linked above to extrapolate and smooth much more in the north-south direction than in the east-west direction. You want to give equivalent weight to points further away along the northing (Y) axis, as you do to points that are close in the easting (X) direction. Describe how well you can do.

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