How to grid unevenly sampled categorical data?

Question

I am looking for a method for gridding categorical data. I have extracted from nautical charts and field sheets a collection of points specifying the nature of the surface of the bottom of the ocean. These data are categorical rather than numerical and they are not regularly nor even randomly sampled. Nautical charts are created to aid in navigation and anchoring; they are not created to map habitat. As such, more soundings are made close to shore where relatively shallow depths can pose a hazard to navigation and where ships to tend to anchor. Further from shore, where the depths are more than adequate for navigation and anchoring is impractical, soundings are made much less frequently.

Has anyone else tried to create gridded substrate maps from nautical charts?

I looked at Thiessen (Vornoi) polygons but the concentration of soundings along shores leads to a fine 'honeycomb' along the shore, large polygons offshore and in between long pie-shaped polygons stretching way offshore. Gridding using nearest neighbour yields pretty much the same result.

I need a way to limit the influence of the shallow, near shore points -- a way to limit those long pie-shaped polygons. In deeper waters I do not expect that the nature of the bottom will be a continuation of the near shore bottom. I started thinking along two lines -- both using depth. One is weighting the choice of the 'nearest' neighbour using the difference in depth between a grid cell and neighbouring points. Another is de-selecting neighbouring points which are more than some specified tolerance different in depth. Or, perhaps rather than a pre-specified tolerance, I could bin depth ranges and then limit the choice of neighbouring points to those in the same depth range or bin.

Any thoughts on how to implement either of these two options?

Since talking with colleagues on other forums, I have been looking at a couple of other approaches. The first involves using a barrier -- the 100m depth contour -- to limit the influence of near shore data. The challenge with this approach is that any of the ESRI interpolation routines that can use barriers are designed to work with continuous data rather than discontinuous data. I could use the barriers to break up the points into shallow near shore and deeper points before creating Thiessen polygons. However, I anticipate rampant edge effects since ArcGIS createsThiessen polygons for rectangular areas not for complex areas.

The second approach -- suggested by several colleagues -- was kriging. I had initially dismissed kriging out of hand because I have only ever considered it for continuous data. The challenge with kriging is that it too is not designed for categorical data. Now, I am looking at cokriging with depth and nature of surface but, any type of kriging is going to involve using integer numerical codes for nature of surface. Afterward the resulting floating point numerical codes will have to be reduced back to the original integer coding. Not pretty.

Can any one suggest other lines to follow? (One could, perhaps, use terrain analysis. For example, slopes steeper than the angle of repose could not be sediments. I am looking for something simpler and, at any rate, I do not have data at sufficient spatial resolution.)

Regards,

Answer 1

The kriging approach, appropriately implemented, is promising.

As a point of departure, look at the "generalized linear geostatistical models" described by Diggle & Ribeiro in Model-based Geostatistics (Springer 2007). The underlying idea is appealing and flexible: a spatial stochastic process (which is spatially continuous) determines the various probabilities of the categories. One uses the observed categories at irregular points to infer statistical properties of that underlying process, including its spatial correlation structure (variogram). Kriging then creates a probability surface consistent with the observations. At that point you can perform geostatistical simulations or you can make maps related to the probabilities (such as maps of the maximum-probability categories, I imagine).

This sounds sophisticated, and it is. However, Diggle and Ribeiro's discussion is fairly accessible--although it's mathematical and assumes some knowledge of statistics, it doesn't rely heavily on either--and most of their techniques are implemented in R packages they describe, geoR and geoRGLM. In fact, it's fair to construe this book as the manual for these packages.

As other threads on this site attest, it's relatively easy to interface R with GIS data (including shapefile and various raster formats), so that's not an issue.