I have an equation that I need to calculate numerically, but I am having doubts about my approach. I also put this question on the Computational Science stack exchange with a bounty.
This is the equation I need to program:
where $\chi$ is a volume mixing ratio in the stratosphere, $\chi_0$ is a volume mixing ratio in the troposphere and G is an age spectrum (a distribution of times it takes an air parcel to travel from troposphere to stratosphere). G is an inverse Gaussian:
where t' is defined as the transit time $\Delta$t divided by $\Gamma$.
The tropospheric volume mixing ratios as a function of time form an almost linear curve. I'll put the curves below (ignore the dashed lines). The time lags are with respect to 2009, i.e. a time lag of 1 corresponds to 2008 etc. Now I need to apply the first equation to these two curves numerically. I have tried numpy.convolve(chi0[::-1], G, mode='same') and variations thereof, but I'm having doubts about the results, so any feedback is greatly appreciated.
So to state it clearly, my question is: How do you convolve the tropospheric curve with the spectrum G, as prescribed by the first equation?
