The Oil Drum: Europe

The convolution of a Gaussian with a Gaussian gives another Gaussian. The sigmas add in quadrature. This is because the product of two Gaussians is a Gaussian and the Fourrier transform of a Gaussian is a Gaussian. It is a special function. I've derived analytically what minimal Gaussian kernels are needed to bring two arbitrarily oriented 2-D Gaussians ellipsiods of differing dimensions into coincidence with the least required distortion to each in an attempt to speed up a program I wrote which derives such kernels for galaxy images obtained with different seeing by means of minimization. The problem there is very degenerate so I wanted to give it a head start. From working on the problem, I can say that you can always take a delta function into another function but after that it gets tricky. Things that look quite narrow sometimes don't have a kernel that takes them into a broader function. But, there is always a pair of kernels that take both into a common function (each other in the trival case) and it is possible to say something about what the best pair looks like.

I think the beginning of the logistics function has a sound basis. If you are making money and you want to make more, you reinvest based on the amount of money you are making. This produces an exponential. The center of the logistics function also has a sound basis: diminishing returns. At some point, putting more effort in isn't getting you anywhere so you put your money somewhere else. I don't know why depletion would look like the logistics function. Perhaps it has to do with wanting to keep some of the people employed who made the whole thing go in the first place. If the coal seam, for example, is running out though, you can only put so many miners at the face. And if an oil well breaks down that is not producing much, you might decide not to fix it. Perhaps depletion is sentimental so it does not get a clear cut off that you might expect.

If you like, I can see if there is a kernel that takes discovery into production via convolution, but convolution is not necessarily the proper method because the time-scale of production probably depends on what has been learned from prior experience. A time varying kernel might be a better representation. It may be possible to take production curves as a function of time and derive a projection that is helpful for estimating the behavior of future or recent discoveries. One catch there though is that the recent curves which are most constraining of future curves may not be complete enough to learn from.

Chris