INTRODUCTION

Twenty-five years ago I attended a series of lectures organized by the University of Houston called ``Petroleum Geology for Geophysicists''. One of the professors, Daniel Busch (if I recall correctly), proposed a data set that would be ``interesting to contour''. He might have said that specialized knowledge of petroleum reservoirs would be helpful. His experience was with very sparse data. I recall it being well logs from Mexico. Of special interest were (and are) sand thicknesses. He cited four wells, each with a measurement:

.                   5          98
.        
.                         x
.        
.                  100          7

The question for the interpreter is, what should be the value of x? Mathematical algorithms generally give a value of x near 53. As a petroleum geologist, Busch was accustomed to visualizing drainage patterns such as rivers with residual sands. For a river running northwesterly, the value of x would be near 6. On the other hand, he said, a paleotopography also commonly contains ridges, so maybe x should be roughly 100.

This example charmed me enough to remember it for 25 years and to relay it to you now, with the hope that we can do something helpful about Busch's problem (that mathematical contouring is not as good as common sense). The solutions that we are most accustomed to are the linear solutions that come from minimizing quadratic forms. Such setups generally give the average value of x near 54 that Busch would like to avoid.

We might wonder if Busch's problem is too hard for any mathematical method. I think we would feel that progress had been made, however, if we uncovered a method that told us this data

.                              98
.                      5
.                         x
.        
.                  100          7

suggests a river while this data

.                    5      
.                            98
.                         x
.        
.                  100          7

suggests a ridge. The whole problem could be more interesting in the presence of more data values. The more remote data values could actually be making the choice of a river channel or a ridge.

The goal is that the result should look more like topography than what we see arising from familiar L₂ methods. The essential aspect of real world topography is that erosion cuts drainage channels.