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	1 <div class="section no-pad-bot" id="index-banner">
	2 <div class="container">
	3
	4 <br><br>
	5
	6 <div class="row">
	7 <div class="col s12 m9 l10">
	8
	9 <h1 class="header center orange-text">Methods</h1>
	10
	11 <div id="introduction" class="section scrollspy">
	12 <h3 class="header blue-text">Introduction</h3>
	13 <p>
	14 Cosmogenic nuclides are typically used to either con…
	15 an exposure age, a burial age, or an erosion rate.
	16 Constraining the landscape history and past erosion …
	17 previously glaciated terrains is, however, notorious…
	18 difficult because it involves a large number of unkn…
	19 This tool uses an approach based on the Markov Chain…
	20 Carlo (MCMC) technique. The model framework current…
	21 incorporates any combination of the following terres…
	22 cosmogenic nuclides (TCNs) <sup>10</sup>Be, <sup>26<…
	23 <sup>14</sup>C, and <sup>21</sup>Ne in order to cons…
	24 two-stage glacial/interglacial history at the site of
	25 sampling.</p>
	26
	27 <p>The MCMC technique is used to simulate TCN concen…
	28 associated with a large number of different
	29 glacial-interglacial histories, including highly var…
	30 glacial and interglacial erosion rates. Based on com…
	31 to measured concentrations, it is possible to determ…
	32 most likely landscape history and associated uncerta…
	33
	34 <p class="flow-text">
	35 The model approach approximates the values
	36 and uncertainties of four output parameters;
	37 interglacial erosion rate (ε<sub>int</su…
	38 glacial erosion rate (ε<sub>gla</sub>), …
	39 last deglaciation (<i>t</i><sub>degla</sub>), an…
	40 climate threshold value in the marine oxygen-iso…
	41 record (δ<sup>18</sup>O<sub>threshold</sub…
	42 the site of sampling.
	43 </p>
	44
	45 <p>In the following we give a basic overview of the …
	46 methods and their application. For a full descriptio…
	47 the open-access publication by <a
	48 href="http://www.sciencedirect.com/science/artic…
	49 et al. (2015)</a>.</p>
	50 </div>
	51
	52 <div id="mcmc" class="section scrollspy">
	53 <h3 class="header blue-text">
	54 Markov-Chain Monte Carlo (MCMC) basics</h3>
	55 <p>The inversion problem of turning observed TCN
	56 concentrations into erosion histories is handled usi…
	57 conventional Metropolis-Hastings MCMC approach. The
	58 model parameters are constrained between fixed model
	59 parameter bounds specified by the user. Erosion rates
	60 (ε<sub>int</sub>, ε<sub>gla</sub>), …
	61 may vary over several orders of magnitude, are tested
	62 with uniform probability across the logarithmic para…
	63 interval. The temporal parameter (<i>t</i><sub>degla…
	64 and climate record threshold value
	65 (δ<sup>18</sup>O<sub>threshold</sub>) are test…
	66 uniform probability across the linear parameter inte…
	67 </p>
	68
	69 <p>When model parameters
	70 (ε<sub>int</sub>, ε<sub>gla</sub>,
	71 <i>t</i><sub>degla</sub>,
	72 δ<sup>18</sup>O<sub>threshold</sub>) are varie…
	73 specified limits, they can be thought of as being or…
	74 axes spanning a coordinate system in four-dimensiona…
	75 Each position in this model space is associated with…
	76 unique set of model parameter values.
	77 </p>
	78
	79 <p>Given a single value of model parameters
	80 (ε<sub>int</sub>, ε<sub>gla</sub>,
	81 <i>t</i><sub>degla</sub>,
	82 δ<sup>18</sup>O<sub>threshold</sub>) within the
	83 specified limits, the TCN concentration after the du…
	84 of e.g. the entire Quaternary period in a sample can…
	85 computed. This <i>forward model</i> describes a poss…
	86 history of exhumation and TCN production in a sample…
	87 as it experiences the variable physical environment …
	88 Quaternary.</p>
	89
	90
	91 <div id="twostage" class="subsection scrollspy">
	92 <h4 class="header blue-text light">
	93 Two-stage glacial-interglacial forward model…
	94 <p>The forward model builds on the assumption of
	95 "two-stage uniformitarianism", meaning that the
	96 processes that operated during the Holocene also
	97 operated during earlier interglacials with compa…
	98 intensity. Likewise, the erosion rate during the…
	99 glacial periods is assumed to be comparable.</p>
	100
	101 <p>The model approach assumes that glacial perio…
	102 characterized by 100% shielding and no exposure,…
	103 would require more than 10 m of ice thickness for
	104 production due to spallation (>50 m for muons…
	105 Interglacial periods are assumed to have been
	106 characterized by 100% exposure and zero shieldin…
	107 production of TCNs takes place during the interg…
	108 while erosion removes the land surface at differ…
	109 rates during the glacials and interglacials.</p>
	110
	111 <p>The forward model switches between glacial and
	112 interglacial state when the selected climate rec…
	113 crosses a threshold value. The provided climate …
	114 are based on a benthic δ<sup>18</sup>O rec…
	115 smoothed by various degrees, implying that clima…
	116 at the site of sample is correlated to the global
	117 state.</p>
	118 </div>
	119
	120 <div id="mcmcwalker" class="subsection scrollspy">
	121 <h4 class="header blue-text light">
	122 What is a MCMC walker?</h4>
	123 <p>
	124 A MCMC walker is in this context a numerical ent…
	125 which sequentially explores the model parameter …
	126 order to obtain the closest match between the fo…
	127 model and the observational dataset of TCNs. Dur…
	128 iteration the walker takes its current position …
	129 space, plugs the parameter value into the forwar…
	130 and evaluates if the output result matches the
	131 observational record better or worse than the ou…
	132 its previous position in model space. If the new…
	133 better matches the observed dataset, it continues
	134 walking in the same direction in model space.
	135 </p>
	136
	137 <p>
	138 Starting at a random place inside the model spac…
	139 burn-in phase of 1000 iterations is first used t…
	140 crude search of the entire model space. The bur…
	141 phase is followed by a similar but more detailed…
	142 local search of the model space, based on the be…
	143 model parameters from the burn-in phase. The we…
	144 least-squared misfit to observed TCN concentrati…
	145 used to evaluate the likelyhood for the combinat…
	146 model parameter values. The MCMC walker continu…
	147 exploring the model space until it is sufficient…
	148 satisfied with the best model parameter estimate…
	149 found.
	150 </p>
	151
	152 <p>
	153 For a given observational data set more than one…
	154 model parameters may produce forward models which
	155 sufficiently satisfy the MCMC walker. In this c…
	156 solution is <i>non-unique</i>. Even worse, a sin…
	157 walker may find an area in model space which see…
	158 is in good correspondence with the observational…
	159 set, but the walker is missing a much better set…
	160 model parameters since they are located somewhere
	161 entirely different in the model space. In order …
	162 mitigate these issues, MCMC inversions are often
	163 performed using several MCMC walkers. The start…
	164 point of each MCMC walker is chosen at random, r…
	165 in unique walks through the model space. If a si…
	166 walker is caught in an area of non-ideal solutio…
	167 chances are that the other walkers will find the…
	168 better model parameters.
	169 </p>
	170
	171 <p>
	172 The computational time depends on the number of …
	173 walkers. When casually trying out the calculator…
	174 recommend using low numbers of MCMC walkers (1 t…
	175 order to obtain fast results and reduce load on …
	176 server. When attempting to produce high-quality…
	177 reliable results, the number of walkers should be
	178 increased (3 to 4).
	179 </p>
	180 </div>
	181 </div>
	182
	183
	184 <div id="citing" class="section scrollspy">
	185 <h3 class="header blue-text">
	186 Citing the MCMC cosmo calculator</h3>
	187
	188 <p>If you use the results generated by this tool in a
	189 scientific publication, please acknowledge this fact…
	190 citing:</p>
	191 <blockquote>
	192 Knudsen, M.F., Egholm, D.L., Jacobsen, B.H., Lar…
	193 J.D., Andersen, J.L., Linge, H.C., 2015.<br>
	194 <b>A multi-nuclide approach to constrain landsca…
	195 past erosion rates in previously glaciated t…
	196 Quaternary Geochronology 30, 100-113,
	197 doi:10.1016/j.quageo.2015.08.004.
	198 </blockquote>
	199
	200 <p>You may use the following BibTeX entry:</p>
	201 <div class="row">
	202 <pre><code class="language-markup col s12"> @…
	203 author = "Knudsen, M. F. and Egholm, D. L. and Jacobsen, B. H.
	204 and Larsen, N. K. and Jansen, J. D. and Andersen, J. L.
	205 and Linge, H. C.",
	206 title = "A multi-nuclide approach to constrain landscape
	207 evolution and past erosion rates in previously glaciated
	208 terrains",
	209 journal = "Quaternary Geochronology",
	210 volume = "30, Part A",
	211 number = "",
	212 pages = "100--113",
	213 year = "2015",
	214 issn = "1871-1014",
	215 doi = "http://dx.doi.org/10.1016/j.quageo.2015.08.004",
	216 }
	217 </code></pre>
	218 </div>
	219 </div>
	220
	221 </div>
	222
	223 <div class="col hide-on-small-only m3 l2">
	224 <div class="toc-wrapper pin-top" style="top: 0px;">
	225 <ul class="section table-of-contents">
	226 <li><a href="#introduction">Introduction</a></li>
	227 <li><a href="#mcmc">MCMC</a></li>
	228 <li><a href="#twostage">Application</a></li>
	229 <li><a href="#mcmcwalker">Walkers</a></li>
	230 <li><a href="#citing">Citing</a></li>
	231 </ul>
	232 </div>
	233 </div>
	234 </div>
	235 </div>
	236 </div>
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