BOE: Gamma PDF Estimate of Equilibrium Climate Sensitivity
Estimating the probability distribution for equilibrium climate sensitivity with a Gamma distribution visually fitted to Meinshausen 2009 Fig 3a” (top). The gamma distribution for the 2 to 4.5C spread for equilibrium climate sensitivity has a cumulative 85% probability.
The gamma distribution is coded as follows:
x = seq(0.1,10,by=0.1)
alpha = 12# width of dist, higher is narrower
beta = alpha/3.2 # shift, higher is right shift
ts_pdf = beta^alpha/gamma(alpha) * x^(alpha-1) * exp(-beta*x)
ts_pdf = ts_pdf/sum(ts_pdf)
The Gamma Distribution was not my first attempt at a fit. My first fit was a Poisson distribution, rescaled and shifted to fit the Meinshausen ECD PDF. The Poisson function for the 2 to 4.5 spread for climate sensitivity also integrates to an 85% probability.
The code for the Poisson fit is as follows:
lambda = 3 # ECS mean = 3
xmax = 14 # long enough tail to go to zero
xx = 0:xmax
xx2 = xx/2 # scale the Poisson back down to 0:7
xxtck = 0:7 # labels for the chart
yy = lambda^xx * exp(-lambda) / factorial(xx) # Poisson
Ts = approx(xx2,yy,n=71) # need more points, so interpolate to dx = 0.1
shift = 17 # shift peak back to ECS = 3
yy2 = rep(0,71)
yy2[shift:71] = Ts$y[1:(71-shift+1)]
Ts$y = yy2 / 5 # shift complete
sum(Ts$y[Ts$x>=2 & Ts$x<=4.5])
The similarity between the two is even more obvious when plotted together. It can’t be accidental, but I haven’t studied the two yet to derive their relationship.