itchange nesting and indentation - cosmo - front and backend for Markov-Chain Monte Carlo inversion of cosmogenic nuclide concentrations Err mx1.adamsgaard.dk 70 hgit clone git://src.adamsgaard.dk/cosmo URL:git://src.adamsgaard.dk/cosmo mx1.adamsgaard.dk 70 1Log /src/log.gph mx1.adamsgaard.dk 70 1Files /src/files.gph mx1.adamsgaard.dk 70 1Refs /src/refs.gph mx1.adamsgaard.dk 70 1README /src/file/README.md.gph mx1.adamsgaard.dk 70 1LICENSE /src/file/LICENSE.gph mx1.adamsgaard.dk 70 i--- Err mx1.adamsgaard.dk 70 1commit 9abb8d8302f7752b7f803bacc0a0a68e3992ee00 /src/commit/9abb8d8302f7752b7f803bacc0a0a68e3992ee00.gph mx1.adamsgaard.dk 70 1parent a53b33ee8911d15bcfd27b48dad8944d0fd86baa /src/commit/a53b33ee8911d15bcfd27b48dad8944d0fd86baa.gph mx1.adamsgaard.dk 70 hAuthor: Anders Damsgaard URL:mailto:anders.damsgaard@geo.au.dk mx1.adamsgaard.dk 70 iDate: Fri, 27 Nov 2015 16:48:47 +0100 Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 ichange nesting and indentation Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 iDiffstat: Err mx1.adamsgaard.dk 70 i M pages/methods.html | 158 ++++++++++++++++--------------- Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i1 file changed, 81 insertions(+), 77 deletions(-) Err mx1.adamsgaard.dk 70 i--- Err mx1.adamsgaard.dk 70 1diff --git a/pages/methods.html b/pages/methods.html /src/file/pages/methods.html.gph mx1.adamsgaard.dk 70 it@@ -87,85 +87,89 @@ Err mx1.adamsgaard.dk 70 i as it experiences the variable physical environment of the Err mx1.adamsgaard.dk 70 i Quaternary.

Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i- Err mx1.adamsgaard.dk 70 i- Err mx1.adamsgaard.dk 70 i-
Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i- Two-stage glacial-interglacial forward model

Err mx1.adamsgaard.dk 70 i-

The forward model builds on the assumption of a Err mx1.adamsgaard.dk 70 i- "two-stage uniformitarianism", meaning that the processes Err mx1.adamsgaard.dk 70 i- that operated during the Holocene also operated during Err mx1.adamsgaard.dk 70 i- earlier interglacials with comparable intensity. Likewise, Err mx1.adamsgaard.dk 70 i- the erosion rate during the past glacial periods is assumed Err mx1.adamsgaard.dk 70 i- to be comparable.

Err mx1.adamsgaard.dk 70 i- Err mx1.adamsgaard.dk 70 i-

The model approach assumes that glacial periods were Err mx1.adamsgaard.dk 70 i- characterized by 100% shielding and no exposure, which would Err mx1.adamsgaard.dk 70 i- require more than 10 m of ice thickness for production due Err mx1.adamsgaard.dk 70 i- to spallation (>50 m for muons). Interglacial periods are Err mx1.adamsgaard.dk 70 i- assumed to have been characterized by 100% exposure and zero Err mx1.adamsgaard.dk 70 i- shielding. The production of TCNs takes place during the Err mx1.adamsgaard.dk 70 i- interglacials, while erosion removes the land surface at Err mx1.adamsgaard.dk 70 i- different rates during the glacials and interglacials.

Err mx1.adamsgaard.dk 70 i-
Err mx1.adamsgaard.dk 70 i- Err mx1.adamsgaard.dk 70 i-
Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i- What is a MCMC walker?

Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i- A MCMC walker is in this context a numerical entity which Err mx1.adamsgaard.dk 70 i- sequentially explores the model parameter space in order to Err mx1.adamsgaard.dk 70 i- obtain the closest match between the forward model and the Err mx1.adamsgaard.dk 70 i- observational dataset of TCNs. During each iteration Err mx1.adamsgaard.dk 70 i- the walker takes its current position in model space, plugs Err mx1.adamsgaard.dk 70 i- the parameter value into the forward model, and Err mx1.adamsgaard.dk 70 i- evaluates if the output result matches the observational Err mx1.adamsgaard.dk 70 i- record better or worse than the output at its previous Err mx1.adamsgaard.dk 70 i- position in model space. If the new results better matches Err mx1.adamsgaard.dk 70 i- the observed dataset, it continues walking in the same Err mx1.adamsgaard.dk 70 i- direction in model space. Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i- Starting at a random place inside the model space, a burn-in Err mx1.adamsgaard.dk 70 i- phase of 1000 iterations is first used to make a crude Err mx1.adamsgaard.dk 70 i- search of the entire model space. The burn-in phase is Err mx1.adamsgaard.dk 70 i- followed by a similar but more detailed and local search of Err mx1.adamsgaard.dk 70 i- the model space, based on the best-fit model parameters from Err mx1.adamsgaard.dk 70 i- the burn-in phase. The weighted least-squared misfit to Err mx1.adamsgaard.dk 70 i- observed TCN concentrations is used to evaluate the Err mx1.adamsgaard.dk 70 i- likelyhood for the combinations of model parameter values. Err mx1.adamsgaard.dk 70 i- The MCMC walker continues exploring the model space until it Err mx1.adamsgaard.dk 70 i- is sufficiently satisfied with the best model parameter Err mx1.adamsgaard.dk 70 i- estimate it has found. Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i- Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i- For a given observational data set more than one set of Err mx1.adamsgaard.dk 70 i- model parameters may produce forward models which Err mx1.adamsgaard.dk 70 i- sufficiently satisfy the MCMC walker. Err mx1.adamsgaard.dk 70 i- In this case the solution is non-unique. Even worse, Err mx1.adamsgaard.dk 70 i- a single MCMC walker may find an area in model space which Err mx1.adamsgaard.dk 70 i- seemingly is in good correspondence with the observational Err mx1.adamsgaard.dk 70 i- data set, but the walker is missing a much better set of Err mx1.adamsgaard.dk 70 i- model parameters since they are located somewhere entirely Err mx1.adamsgaard.dk 70 i- different in the model space. In order to mitigate these Err mx1.adamsgaard.dk 70 i- issues, MCMC inversions are often performed using several Err mx1.adamsgaard.dk 70 i- MCMC walkers. The starting point of each MCMC walker is Err mx1.adamsgaard.dk 70 i- chosen at random, resulting in unique walks through the Err mx1.adamsgaard.dk 70 i- model space. If a single walker is caught in an area of Err mx1.adamsgaard.dk 70 i- non-ideal solutions, chances are that the other walkers will Err mx1.adamsgaard.dk 70 i- find the area of better model parameters. Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i+
Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Two-stage glacial-interglacial forward model

Err mx1.adamsgaard.dk 70 i+

The forward model builds on the assumption of a Err mx1.adamsgaard.dk 70 i+ "two-stage uniformitarianism", meaning that the Err mx1.adamsgaard.dk 70 i+ processes that operated during the Holocene also Err mx1.adamsgaard.dk 70 i+ operated during earlier interglacials with comparable Err mx1.adamsgaard.dk 70 i+ intensity. Likewise, the erosion rate during the past Err mx1.adamsgaard.dk 70 i+ glacial periods is assumed to be comparable.

Err mx1.adamsgaard.dk 70 i+ Err mx1.adamsgaard.dk 70 i+

The model approach assumes that glacial periods were Err mx1.adamsgaard.dk 70 i+ characterized by 100% shielding and no exposure, which Err mx1.adamsgaard.dk 70 i+ would require more than 10 m of ice thickness for Err mx1.adamsgaard.dk 70 i+ production due to spallation (>50 m for muons). Err mx1.adamsgaard.dk 70 i+ Interglacial periods are assumed to have been Err mx1.adamsgaard.dk 70 i+ characterized by 100% exposure and zero shielding. The Err mx1.adamsgaard.dk 70 i+ production of TCNs takes place during the interglacials, Err mx1.adamsgaard.dk 70 i+ while erosion removes the land surface at different Err mx1.adamsgaard.dk 70 i+ rates during the glacials and interglacials.

Err mx1.adamsgaard.dk 70 i+
Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i- The computational time depends on the number of MCMC Err mx1.adamsgaard.dk 70 i- walkers. When casually trying out the calculator we Err mx1.adamsgaard.dk 70 i- recommend using low numbers of MCMC walkers (1 to 2) in Err mx1.adamsgaard.dk 70 i- order to obtain fast results and reduce load on the server. Err mx1.adamsgaard.dk 70 i- When attempting to produce high-quality and reliable Err mx1.adamsgaard.dk 70 i- results, the number of walkers should be increased (3 to 4). Err mx1.adamsgaard.dk 70 i-

Err mx1.adamsgaard.dk 70 i+
Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ What is a MCMC walker?

Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ A MCMC walker is in this context a numerical entity Err mx1.adamsgaard.dk 70 i+ which sequentially explores the model parameter space in Err mx1.adamsgaard.dk 70 i+ order to obtain the closest match between the forward Err mx1.adamsgaard.dk 70 i+ model and the observational dataset of TCNs. During each Err mx1.adamsgaard.dk 70 i+ iteration the walker takes its current position in model Err mx1.adamsgaard.dk 70 i+ space, plugs the parameter value into the forward model, Err mx1.adamsgaard.dk 70 i+ and evaluates if the output result matches the Err mx1.adamsgaard.dk 70 i+ observational record better or worse than the output at Err mx1.adamsgaard.dk 70 i+ its previous position in model space. If the new results Err mx1.adamsgaard.dk 70 i+ better matches the observed dataset, it continues Err mx1.adamsgaard.dk 70 i+ walking in the same direction in model space. Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Starting at a random place inside the model space, a Err mx1.adamsgaard.dk 70 i+ burn-in phase of 1000 iterations is first used to make a Err mx1.adamsgaard.dk 70 i+ crude search of the entire model space. The burn-in Err mx1.adamsgaard.dk 70 i+ phase is followed by a similar but more detailed and Err mx1.adamsgaard.dk 70 i+ local search of the model space, based on the best-fit Err mx1.adamsgaard.dk 70 i+ model parameters from the burn-in phase. The weighted Err mx1.adamsgaard.dk 70 i+ least-squared misfit to observed TCN concentrations is Err mx1.adamsgaard.dk 70 i+ used to evaluate the likelyhood for the combinations of Err mx1.adamsgaard.dk 70 i+ model parameter values. The MCMC walker continues Err mx1.adamsgaard.dk 70 i+ exploring the model space until it is sufficiently Err mx1.adamsgaard.dk 70 i+ satisfied with the best model parameter estimate it has Err mx1.adamsgaard.dk 70 i+ found. Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ For a given observational data set more than one set of Err mx1.adamsgaard.dk 70 i+ model parameters may produce forward models which Err mx1.adamsgaard.dk 70 i+ sufficiently satisfy the MCMC walker. In this case the Err mx1.adamsgaard.dk 70 i+ solution is non-unique. Even worse, a single MCMC Err mx1.adamsgaard.dk 70 i+ walker may find an area in model space which seemingly Err mx1.adamsgaard.dk 70 i+ is in good correspondence with the observational data Err mx1.adamsgaard.dk 70 i+ set, but the walker is missing a much better set of Err mx1.adamsgaard.dk 70 i+ model parameters since they are located somewhere Err mx1.adamsgaard.dk 70 i+ entirely different in the model space. In order to Err mx1.adamsgaard.dk 70 i+ mitigate these issues, MCMC inversions are often Err mx1.adamsgaard.dk 70 i+ performed using several MCMC walkers. The starting Err mx1.adamsgaard.dk 70 i+ point of each MCMC walker is chosen at random, resulting Err mx1.adamsgaard.dk 70 i+ in unique walks through the model space. If a single Err mx1.adamsgaard.dk 70 i+ walker is caught in an area of non-ideal solutions, Err mx1.adamsgaard.dk 70 i+ chances are that the other walkers will find the area of Err mx1.adamsgaard.dk 70 i+ better model parameters. Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+ The computational time depends on the number of MCMC Err mx1.adamsgaard.dk 70 i+ walkers. When casually trying out the calculator we Err mx1.adamsgaard.dk 70 i+ recommend using low numbers of MCMC walkers (1 to 2) in Err mx1.adamsgaard.dk 70 i+ order to obtain fast results and reduce load on the Err mx1.adamsgaard.dk 70 i+ server. When attempting to produce high-quality and Err mx1.adamsgaard.dk 70 i+ reliable results, the number of walkers should be Err mx1.adamsgaard.dk 70 i+ increased (3 to 4). Err mx1.adamsgaard.dk 70 i+

Err mx1.adamsgaard.dk 70 i+
Err mx1.adamsgaard.dk 70 i
Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 i Err mx1.adamsgaard.dk 70 .