Q:"why look for your lost keys only under the streetlamp?"
A:"cuz that's where the light is"
I found a bit of mathematical convenience that perhaps can add a bit of clarity to the understanding of the Oil Shock Model. In the past I have used a few other devices to get the point across, including an electrical circuit analogy and the use of gamma distributions. The new construction I present came up in a previous comment at TOD.
Consider a hypothetical situation that a discovery profile fits a Gaussian density profile in time. Then consider that each shift in the production history also follows a Gaussian mean with attached variance. It follows (in keeping with the formulation of the Shock Model) that a convolution of a Gaussian density function with a Gaussian results in another Gaussian (proof let to the reader1). The resultant width adds in quadrature and the Gaussian shifts by the relative offset of the two curves. This essentially means that after N repeated convolutions the peak shifts by N mean values and the width broadens by the square root of the sum of the squares of the N standard deviations. Both the Shock Model and this hypothetical analysis used the concept of the mathematical convolution to demonstrate how the oil production curve shifts in time from the initial discovery profile. In the global situation this shift often manifests itself by latency of dozens of years (the current shift runs at +40 years).
The repeated convolutions also cause the initial discovery profile to broaden due to probabilistic considerations, essentially explainable by the uncertainties in when production and other preceding phases actually start on a given discovered region. The only way that the profile can sharpen after discovery occurs by increases in the extraction rate as evidenced by demand, technological advancements, or other various production shocks. These non-stochastic properties can alone counteract the relentless advance of entropy.
the curves above correspond to the equations below where w=1 and dt=5 years:
The use of Gaussians to describe the convolutions allow a closed-form solution to the result and at each stage. For background on how the central limit theorem also plays into this see here. Note however, that the closed form solution only holds if the Gausians have negative time histories, which unfortunately breaks the time causality of discovery and production. But like I said, if we use this simply as a means to a better understanding in how the continuous model manifests itself, by truncating the negative tails and eyeballing the curves, we can essentially live with the approximation. It gives us a intuitive shorthand in understanding how the latencies add up and how the resultant peak can broaden given the underlying density functions.P0=exp(-(t)2/2/w2)/sqrt(w2)
-- Discovery phase centered at 0, width = wP1=exp(-(t-dt)2/2/(w2+w2))/sqrt(w2+w2)
-- Fallow phase, mean latency = dt, variance = w2P2=exp(-(t-2*dt)2/2/(w2+w2+w2))/sqrt(w2+w2+w2)
-- Construction phase, mean latency = dt, variance = w2P3=exp(-(t-3*dt)2/2/(w2+w2+w2+w2))/sqrt(w2+w2+w2+w2)
-- Maturation phase, mean latency = dt, variance = w2P4=exp(-(t-4*dt)2/2/(w2+w2+w2+w2+w2))/sqrt(w2+w2+w2+w2+w2)
-- Production phase, draw-down time = dt, variance = w2
1 For an alternative explanation of how to derive the Gaussian convolution identity, look up the concept of Fourier transforms here. The derivation works out simply if we use the identity that a convolution in the time domain corresponds to a multiplication in the frequency domain, and that the Fourier transform of a Gaussian results in a Gaussian.
