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I have a question about solving a particular graphical problem.

This is a picture of a Neutron-Density cross-plot: enter image description here

It's a little bit confusing as plots go, so allow me to try to explain the salient details.

This chart helps in interpreting well log measurements. A sedimentary rock may have been measured using a Neutron tool and a Bulk Density tool. Also superimposed on this chart are blue lines that represent ideal cases, and each line is numbered with porosity values. If a particular measurement (i.e., a particular x,y coordinate) were to fall along the dolomite line, for example, and we know that the rock is 100% dolomite, the number on that dolomite line is a good estimate of the porosity of the rock. Likewise for a measurement on the calcite line that was taken in 100% calcite. However, if the measurement does not fall directly on one of the blue lines, or if the unit is some unknown mix of calcite and dolomite, then the porosity is estimated by the "iso-porosity" lines (plotted in red). So, porosity would be estmiated by which iso-porosity line that particular coordinate falls on.

So, my question is: given any (x,y) coordinate, is there a mathematical way to know which "iso-porosity" line that point falls on (keep in mind, there can be an infinite number of iso-porosity lines; I've only drawn on the iso-porosity lines of integers)?

Ultimately I think what I'm looking for is how to understand and describe mathematically how the red lines change across the graph. They are just a series of lines, each with a different slope and intercept. If I know the slope and intercept of each line, can I then come up with an equation or series of equations that would describe the character of how the isoporosity lines change across the graph? If so, can I then figure out what the isoporosity line is for any given coordinate? Or is there a different way to approach this problem?

I want a function for isoporosity, where the value of isoporosity is a function of the slope and intercept of each of the red lines on the graph. Then, if I have any x,y coordinate, maybe I could solve to determine which slope and intercept (of the isoporosity function) the point falls on. Is this approach feasible, and if so how do I do it?

EDIT: Ok, I've been thinking a lot about this problem and I am wondering if the approach to take would be to think of the isoporosity lines as z-values. If I had an equation for the surface (f(x,y)), I could solve for z if I know x and y. How do I develop an equation of an irregular surface?

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  • $\begingroup$ I think this is difficult to answer because although there is a lot of detail in your question, we don't know how the isoporosity lines are calculated. Also, the tag "graph theory" is slightly wrong for this. Have you considered asking on the physics stackexchange or a similar forum? $\endgroup$ Commented Nov 23, 2014 at 23:47
  • $\begingroup$ The isoporosity lines are not calculated; the blue lines are empirical, and the iso-porosity lines are simply straight lines that connect porosities of equal values from one blue line to the next. I thought Mathematics would be a good place to ask this, because I'm less interested in the physics of the measurements than I am in getting at a way to determine mathematically which red line a coordinate falls on. What tags other than "graph theory" would you suggest? $\endgroup$
    – dagrha
    Commented Nov 23, 2014 at 23:57
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    $\begingroup$ You're trying to interpolate the isoporisity lines. As for tags, I suggest before adding tags at random you try to find out what they mean, otherwise you'll attract downvotes for being careless. $\endgroup$ Commented Nov 24, 2014 at 0:27
  • $\begingroup$ With all due respect, it's really not strictly an "interpolation" problem. Were a point to fall directly on one of the red lines, there would not be any interpolation involved. It's really more a question about how to understand and describe mathematically how the red lines change across the graph. If I can describe that, I hope that I can then use that information to tell which isoporosity line intersects any given point. $\endgroup$
    – dagrha
    Commented Nov 24, 2014 at 5:06
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    $\begingroup$ I didn't downvote so I have no idea, but the category being wrong might have been the reason. $\endgroup$ Commented Nov 24, 2014 at 7:27

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