Algorithm S1.

Metropolis Hastings algorithm applied to the parameter estimation problem described here. The algorithm is repeated for $N_{It}$ iterations, of which some initial subset is discarded as burn-in. At each iteration, a set of values for each parameter type (e.g., OH concentration, interhemispheric exchange rate, etc.) is proposed jointly (e.g., for all years). The scalars $U$ and $n$ are pseudorandom numbers chosen at each iteration from a uniform [$\mathcal{U}(\mathrm{0,1})$] and Gaussian [$\mathcal{N}(\mathrm{0,1})$] distribution, respectively

Set ${𝜽}_0⟵{𝜽}_{prior}$

Forward model run to calculate ${𝐲}_{model}({\theta }_0,{\mathrm{\varphi }}_0)$

for $i\in [1,N_{It}]$ do

for $pi\in [1,N_{param}]$ do

Propose new hyperparameter state for parameter $pi$ PDF if

required: ${\mathrm{\varphi }}_p⟵{\mathrm{\varphi }}_{i-1}+n_1\mathrm{\Delta }\mathrm{\varphi }$

Propose new parameter state for parameter $pi$: ${\theta }_p⟵{\theta }_{i-1}+$

$n_2\mathrm{\Delta }\theta$

Run model to determine ${𝐲}_{model}({\theta }_p,{\mathrm{\varphi }}_p)$

$A⟵ln(\rho (𝐲({\theta }_p,{\mathrm{\varphi }}_p)|{\theta }_p,{\mathrm{\varphi }}_p))-ln(\rho (𝐲({\theta }_{i-1},{\mathrm{\varphi }}_{i-1})|{\theta }_{i-1},{\mathrm{\varphi }}_{i-1}))+$

$ln(\rho ({\theta }_p,{\mathrm{\varphi }}_p))-ln(\rho ({\theta }_{i-1},{\mathrm{\varphi }}_{i-1}))$

if $ln(U)<A$, then

Accept proposed state ${\theta }_i⟵{\theta }_p$

else

Reject proposed state ${\theta }_i⟵{\theta }_{i-1}$