I was assuming 50 % reduction in international trade, and 50 % of that reduction being caused by climatic effects, so only 25 % (= 0.5^2) caused by climatic effects. I have changed “50 % of it” to “50 % of this loss” in my original reply to clarify.
That makes sense. Thanks for putting the figure in!
I guess famine deaths due to the climatic effects are described by a logistic function, which is a strictly increasing function, so I agree with the above. However, I guess the increase will be pretty small for low levels of soot.
If it were linear starting at 10.5 Tg and going to 22.1 Tg, versus linear starting at 0 Tg and going to 22 Tg, then I think the integral (impact) would be about four times as much. But I agree if you are going linear from 10.4 Tg versus logistic from 0 Tg, the difference would not be as large. But it still could be a factor of two or three, so I think it’s good to run a sensitivity case.
If it were linear starting at 10.5 Tg and going to 22.1 Tg, versus linear starting at 0 Tg and going to 22 Tg, then I think the integral (impact) would be about four times as much.
You are right about that integral, but I do think that is the relevant BOTEC. What we care about is the mean death rate (for a given input soot distribition), not its integral. For example, for a uniform soot distribution ranging from 0 to 37.4 Tg (= 2*18.7), whose mean matches mine of 18.7 Tg[1], the middle points of the linear parts would be:
If the linear part started at 10.5 Tg, 7.27 % (= ((10.5 + 37.4)/2 − 10.5)/(18.7 − 10.5)*0.0443).
If the linear part started at 0 Tg, 10.1 % (= ((0 + 37.4)/2 − 0)/(18.7 − 10.5)*0.0443).
So the mean death rates would be:
If the linear part started at 10.5 Tg, 5.23 % (= (10.5*0 + (37.4 − 10.5)*0.0727)/37.4).
If the linear part started at 0 Tg, 10.1 %.
This suggests famine deaths due to the climatic effects would be 1.93 (= 0.101/0.0523) times as large if the linear part started at 0 Tg.
Another way of running the BOTEC is considering an effective soot level, equal to the soot level minus the value at which the linear part starts. My effective soot level is 8.20 Tg (= 18.7 − 10.5), whereas it would be 18.7 Tg if the linear part started at 0 Tg, which suggests deaths would be 1.78 (= 18.7/10.5) times as large in the latter case. Using a logistic function instead of a linear one, I think the factor would be quite close to 1.
But I agree if you are going linear from 10.4 Tg versus logistic from 0 Tg, the difference would not be as large. But it still could be a factor of two or three, so I think it’s good to run a sensitivity case.
The challenge here is that the logistic function f(x) = a + b/(1 + e^(-k(x—x_0))) has 4 parameters, but I only have 3 conditions, f(0) = 0, f(18.7) = 0.0443, f(+inf) = 1. I think this means I could define the 4th condition such that the logistic function stays near 0 until 10.5 Tg.
Ideally, I would define the logistic function for f(0) = 0 and f(+inf) = 1, but then finding its parameters fitting it to the 16, 27, 37, 47 and 150 Tg cases of Xia 2022 for international food trade, all livestock grain fed to humans, and no household food waste. Then I would use f(18.7) as the death rate. Even better, I would get a distribution for the soot, generate N samples (x_1, x_2, …, and x_N), and then use (f(x_1) + f(x_2) + … + f(x_N))/N as the death rate.
That makes sense. Thanks for putting the figure in!
If it were linear starting at 10.5 Tg and going to 22.1 Tg, versus linear starting at 0 Tg and going to 22 Tg, then I think the integral (impact) would be about four times as much. But I agree if you are going linear from 10.4 Tg versus logistic from 0 Tg, the difference would not be as large. But it still could be a factor of two or three, so I think it’s good to run a sensitivity case.
You are right about that integral, but I do think that is the relevant BOTEC. What we care about is the mean death rate (for a given input soot distribition), not its integral. For example, for a uniform soot distribution ranging from 0 to 37.4 Tg (= 2*18.7), whose mean matches mine of 18.7 Tg[1], the middle points of the linear parts would be:
If the linear part started at 10.5 Tg, 7.27 % (= ((10.5 + 37.4)/2 − 10.5)/(18.7 − 10.5)*0.0443).
If the linear part started at 0 Tg, 10.1 % (= ((0 + 37.4)/2 − 0)/(18.7 − 10.5)*0.0443).
So the mean death rates would be:
If the linear part started at 10.5 Tg, 5.23 % (= (10.5*0 + (37.4 − 10.5)*0.0727)/37.4).
If the linear part started at 0 Tg, 10.1 %.
This suggests famine deaths due to the climatic effects would be 1.93 (= 0.101/0.0523) times as large if the linear part started at 0 Tg.
Another way of running the BOTEC is considering an effective soot level, equal to the soot level minus the value at which the linear part starts. My effective soot level is 8.20 Tg (= 18.7 − 10.5), whereas it would be 18.7 Tg if the linear part started at 0 Tg, which suggests deaths would be 1.78 (= 18.7/10.5) times as large in the latter case. Using a logistic function instead of a linear one, I think the factor would be quite close to 1.
The challenge here is that the logistic function f(x) = a + b/(1 + e^(-k(x—x_0))) has 4 parameters, but I only have 3 conditions, f(0) = 0, f(18.7) = 0.0443, f(+inf) = 1. I think this means I could define the 4th condition such that the logistic function stays near 0 until 10.5 Tg.
Ideally, I would define the logistic function for f(0) = 0 and f(+inf) = 1, but then finding its parameters fitting it to the 16, 27, 37, 47 and 150 Tg cases of Xia 2022 for international food trade, all livestock grain fed to humans, and no household food waste. Then I would use f(18.7) as the death rate. Even better, I would get a distribution for the soot, generate N samples (x_1, x_2, …, and x_N), and then use (f(x_1) + f(x_2) + … + f(x_N))/N as the death rate.
18.7 Tg is the mean stratospheric soot until the end of year 2 corresponding to an initial injection of 22.1 Tg.