Contouring surfaces

Reading: Computer contouring, in Davis, 1986, Statistics and Data Analysis in Geology, Wiley, p. 353-383. Available as a pdf on Canvas.

Why contour?

A primary reason is to visualize an actual or theoretical surface, and see patterns or find anomalies, to understand. Literature exists on learning 'styles' that people naturally have and can develop. A visual learning style is one of the most common and contour maps feed into this ability. The literature on scientific visualization is a significant topic for good reason. However, a word of caution, visualization involves algorithms which can very much change the outcome and appearance of the product and thus influence subsequent interpretations.

Sometimes there is a more practical reason to contour. One may want to interpolate in between points sampled on a surface to estimate the value for that specific point. For example, the desire may be to estimate drilling depth to a given horizon at a new locality based on nearby drill holes, or to estimate the volume of oil in a dome. Sometimes the contour surface may be input for other modeling (e.g. contour maps of groundwater tables yields flow nets).

Contour maps are a very common product of the environmental, oil and mineral industry. If you aren't producing them in your career, you will at the very least be using them a lot!

Finally, contour maps can also be very aesthetically pleasing to create and to see.

Coordinate systems and map projections.

A very basic form of data input for surface modeling is typically - x, y, z, where x and y represent geographic position and z is some value of interest. If z is elvation then this is a topographic representation. When there are geographic coordinates involved the coordinate system is important. The two most commonly used geographic coordinate systems used are:

• latitude and longitude: These are positions described by angular relationships with the earth's rotation axis as a fundamental line of reference, and typically involve considerable distortion of lengths and areas, especially for east-west lines. Note that in order to make things easier you will often want to transform degrees° minutes' and seconds" into decimal degrees. This is easy to do. Divide the minutes by 60 and the seconds by 360 and add both to the degrees.
• UTM (Universal Transverse Mercator): if you want to minimize distortion involved with representing a curved surface on a flat page, the UTM system is often used. A projection is the mathematical algorithm by which points on the curved earth's surface are mapped onto a flat sheet of paper. One has to quantify the actual earth's shape as an ellipsoid in order to do this. The geodetic datum is an attempt to describe its shape with much greater precision.The earth is also broken up into UTM zones by longitude and latitude, and for each zone a position is described by an easting and northing. The easting is the number of meters east of the lower left reference corner of that particular UTM zone, and the northing is the number of meters north of the same reference corner. Note that this is a conventional x, y coordinate system, with eastings plotted along the x axis, and northings along the y axis. Below is a map from the USGS site on UTM showing the zones.
• The geodetic datum model for the earth is important when positional precision is required. NAD83 is a commonly used geodatum associated with North America. It differs a bit from WGS84, a widely used ellipsoid model for the earth. The most important thing is to know what coordinate system and geodetic datum model your data is in. If you plot longitude and latitudes with different projections you will get misplots. This misplots can easily be 100s of meters, and so can be significant. If you need to register two data sets of positions with each other, then they must use the same geodetic datum and projection system.
• Most GPS units can be set to give either latitude and longitude or UTM. I find it often advantageous to record information initially in the field latitude and longitude. The GPS system is based on the ellipsoid model in WGS84. This is also the one that Google Earth uses.

Types of 'earth science' surfaces often contoured?

There are many. Below is a partial list.

• topographic.
• subsurface surfaces:
• groundwater.
• fault surface, fault slip.
• stratigraphic contact.
• isopachs (thicknesses).
• chemical concentration in soil or water (e.g. think contaminant plumes).
• concentration of a discrete entity in a sampling area (e.g. of fracture density, population density, orientations in stereonet space).
• geophysical:
• magnetic anomaly maps.
• gravity anomaly maps.
• heat flow.
• a large variety of possibilities.
• A contour map is a model, an approximation of the real surface. Always remember this!

Examples.

Above is an example of a strip across the Cedar Creek 7.5' USGS topographic quad, showing both contour lines and an underlying shaded relief map. It was generated from a DEM file (Digital Elevation Model), which consists of x, y, z vlues on topography. What basic information is missing from this image that would help make it a more usable map? What patterns do you see in this topography and why do you think they exist?

Puget Sound Aeromagnetic Maps and Data By Richard J. Blakely1, Ray E. Wells1, and Craig S. Weaver2, U.S. Geological Survey Open-File Report 99-514 Version 1.0, 1999.
Note that this USGS map does not show the typical contour lines as seen on many maps of surfaces. Instead it is a shaded relief map. We will discuss these more. With the advent of cheap computing power and software these have become much more common. What patterns do you see in this map, and why do they exist?
One thing to realize with some of these maps is that there is shading based on an illumination point. This point can be changed and the apperance of the map will be somewhat changed. This can be especially important when it comes to picking up lineaments. Linear features aligned with the illumination will be visually less apparent than those at a high angle that have a distinct shadow, which of course can influence subjective visual interpretation.

Aeromagnetic contour map of Georgia. Note the very striking linear pattern in the middle of the state - this reflects various faults and geologic belts in the hinterland of Southern Appalachian geology. Exposures or more limited here due to soil and plant cover, so such maps can be of help in mapping. The use of contour and shaded relief maps to image geophysical data is standard. Image source USGS site: http://pubs.usgs.gov/of/2001/of01-106/

Contour map of Mercury concentrations in Long Island Sound. What conclusions can you draw from this map pattern. How well constrained do you think various aspects of the pattern are (how would additional samples change the pattern? Image source USGS site: http://pubs.usgs.gov/of/2000/of00-304/htmldocs/chap07/index.htm

Contouring algorithms.

Contouring by hand can work well, but is much more subjective. This is one reason computer generated maps are now preferred. However, expert knowledge can help quite a bit in producing a contour map if you know the character of the surface. This might be considered the 'art' behind the product. A simple example is rounded vs. angular geometries for a folded surface. If you know the fold style from observation and/or experience then you can better complete the contours given the discretion one has between control points. Such expert knowledge helps you draw a better contour map with less data. A better solution is to have better control, i.e. more data points, but data costs!! It is also quite important to use your expert knowledge about what is being mapped to evaluate the maps that computer programs generate for you.

Surface trend analysis and mathematical surfaces: This is similar to fitting a line in 2-D graph space - a plane or some more complex curved surface can be fit in 3-D. One advantage of such an analysis is that the surface can then be very efficiently represented or captured - as an equation. This is a link to a quick exploration of modeling surfaces in Excel as a combination of continuous functions and random fluctuations. One can also get useful information by taking derivatives of the surfaces. Such derivatives will map slopes and gradients of change in slope. This is one of the options in Surfer. It can be especially useful if you want to look for anomalies. For example a regional slope can be modeled, and then the difference between that and observed values is the residual, and can be thought of as an anomaly. This is a thought path often used in gravity modeling in geophysics.

Contouring algorithms: This is a crucial consideration! Different algorithms can produce very different results. The fewer control points you have the greater the difference can be.

• Triangulation and interpolation between points is a simple approach, and one that often guides hand contouring. The point where a contour line should intersect a line between two constraining data points is interpolated in some manner. Then lines can be drawn connecting the identified points at a particular contour level. A linear interpolation between two points assumes a local planar character, and thus the model surface is assume to be faceted like a crystal. This is a first approximation, but the resulting contour lines are often not realistic looking.
• Gridding and distance weighting functions: Gridding is where values are estimated/computed for a grid of points, given primary data, which is not evenly distributed (the typical case). This makes subsequent calculation of contour line positions, and/or production of DEMs and digital relief images, much easier. Gridding algorithms can be very complex, but basically the value of a grid point is computed as a function of the nearest neighbor control points, with closer points weighted more. Obviously, the farther away a data control point is from a given grid point, the less influence it should have in the computation for the surface value for that point. Kriging is a 'magic' word often used, but in many situations is argued to be one of the best approaches. Pay close attention to this in your reading. You will explore the difference different contouring algorithms make. Once again, the smaller your data set, the more important this is.
• Link to direct comparison of different gridding techniques - http://www.spatialanalysisonline.com/output/html/Griddingandinterpolationmethods.html
• More detailed looking at kriging - http://oilandgastraining.org/data/gl61/G3921.asp?Code=23365

Wire frames and other forms of visualization.

In addition to the classic contour lines, many other ways of representing or visualizing a surface can now be done easily. Wire frame diagrams attempt to give a realistic rendering of the surface from a defined oblique perspective. Shaded relief maps typically provide a birds view perspective of a obliquely illuminated surface with a color scheme that reflects z values. Finally, animations known as fly-through are also popular. Some examples are given below.

• One example of utility of transect (wire frame) models - Justin Covey's thesis - check out the transects.
• DEMs and shaded relief maps. Digital Elevation Models are computer files that list elevation values for a spatial grid with a given spacing. These are becoming more and more common. Surfer can produce shaded relief maps from a gridded surface. It can also work with shaded relief images. DEMs for Nebraska 7.4 minute USGS topographic quads are available from the Department of Natural Resources, both with a 30 m and 10 m grid spacing.
• fly-through animations (some examples). This is really good techno-geek stuff, and beyond the scope of this course.

Image model multiple subsurface geologic layers for basin analysis and petroleum exploration purposes in Alaska. Image source: http://energy.er.usgs.gov/gg/research/modeling.html

Contouring software.

Software platforms:

• Surfer is a commercial software program that works very well for modeling surfaces. Its major draw back is the cost, but we are fortunate to have access to it. This is the program we will use, and it is often used in industry.
• Surface 3. This is contouring software for the Mac available from the Kansas Geological Survey. It also has some nice online description of how the program works, discussing different gridding options and some of the theory. However, it has not been supported with time and works less and less well (or not at all ) with OS 10 versions.
• ArcMap also had the ability to contour data, which we will introduce you to later. The contouring routines are to be found in ArcToolbox. This is a topic beyond the scope of this course, but there are a variety of help options available within ArcGIS and out on the web that you can use to teach yourself how to do this.
• There are a variety of freeware contouring programs available that can be readily found through the Google search engine. I am unacquainted with them.

• Exercise 5: Producing contour maps of geoscience data.

Data sets to play and learn with.

Possible data sources:

• Quality-Assessed Agrichemical Contaminant Database for Nebraska Ground Water.
• Utah Well and Spring Database.
• USGS earthquake catalog and subducting slabs: If you select by depth in the advanced options section the search will only return those earthquakes deeper than what is specified. A value of 70 km should only select subducting slab related earthquakes. If you have the results returned as csv the position is latitude and longitude, which will involve spatial distortion. There are a variety of ways one can change latitude and longitude to UTM (see above). One simple work around is to have the USGS search site return a kml file, which you can open in Google Earth. If you change Google Earth so that it gives UTM position instead of latitude and longitude, then when you mouse over an earthquake dot, it shows you the earthquake size and depth. You can then insert the UTM and magnitude values into an Excel sheet. Naturally you would not want to do this with a large number of points, but for the purposes of this exercise 30 or so points should allow you to contour the earthquake depth values. Thinking about the gridding algorithm to use in this particular case will be useful. Watch to see if the data is in different UTM zones.

Google Earth provides some nice opportunities for generating your own data sets. Some examples are provided below. 30-40 points can be adequate enough to make a telling contour map for the purposes of this exercise. You can set Google Earth so that instead of providing latitude and longitude of a cursor point it provides the UTM position. This is very useful because then x and y are in meters, and the spatial distortion inherent with latitude and longitude is avoided. Simply contouring the existing landscape would not be very interesting, but there are a variety of situations where you can contour something else by obtaining x-y points at its surface expression. Of course your results will only be good as the underlying Google Earth DEM accuracy permits.

Groundwater tables in the Nebraska Sand Hills: Usually the groundwater table is not visible, but is in the subsurface. However, in the Nebraska Sand Hills lakes are directly connected to the groundwater table. By collecting the center position and elevation of these lakes one can then contour and model what the regional groundwater slope is. The actual pattern will be more complex as the water table will slope up a bit in the surround dunes, but the overall groundwater flow direction will be down the regional slope. There are three distinct areas of such lakes in the Sand Hills.

A similar exercise can be done for parts of Iceland that are dotted by lakes. One are that works well is around 65.88° and -22.08°.

Tilted Layers: The Sierra Nevada Mountains in California have a very asymmetric character from west to east. In the area of 39.68° and -121.68° one can see gently tilted layers dissected by recent erosion. One can pick out a fairly distinctive traceable sedimentary horizon in Google Earth and sample points along it on either side of adjacent valleys where it is exposed. Contouring this provides insight into the regional orientation of these layers, which in turn can provide insight into these mountains. A bit of web research will identify the age of these layers.

In general in more mountainous areas where layers are tilted and folded it can be possible to follow a distinctive layer or a distinctive stratigraphic contact up and down across valleys and ridges. By capturing x, y and z points on the surface trace of that distinctive feature one can then make a structure contour map on the layer or surface and better understand the structural geometry. The strike and dip of approximately planar portions can then be calculated.

Contouring dissected remnant geomorphic surfaces: In some places, such as just east of Broken Bow headward erosion has cut back into an older geomorphic surface, leaving bits of the surface preserved between the various drainage divides. By obtaining x, y, z points on the preserved sections of the remnant surface one can contour to build a model of what the morphology of the older surface might have been.