tmethods.html - cosmo - front and backend for Markov-Chain Monte Carlo inversion of cosmogenic nuclide concentrations
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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 constrain
15 an exposure age, a burial age, or an erosion rate.
16 Constraining the landscape history and past erosion rates in
17 previously glaciated terrains is, however, notoriously
18 difficult because it involves a large number of unknowns.
19 This tool uses an approach based on the Markov Chain Monte
20 Carlo (MCMC) technique. The model framework currently
21 incorporates any combination of the following terrestrial
22 cosmogenic nuclides (TCNs) <sup>10</sup>Be, <sup>26</sup>Al,
23 <sup>14</sup>C, and <sup>21</sup>Ne in order to constrain a
24 two-stage glacial/interglacial history at the site of
25 sampling.</p>
26
27 <p>The MCMC technique is used to simulate TCN concentrations
28 associated with a large number of different
29 glacial-interglacial histories, including highly varying
30 glacial and interglacial erosion rates. Based on comparisons
31 to measured concentrations, it is possible to determine the
32 most likely landscape history and associated uncertainties.
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</sub>),
38 glacial erosion rate (ε<sub>gla</sub>), time of
39 last deglaciation (<i>t</i><sub>degla</sub>), and the
40 climate threshold value in the marine oxygen-isotope
41 record (δ<sup>18</sup>O<sub>threshold</sub>) at
42 the site of sampling.
43 </p>
44
45 <p>In the following we give a basic overview of the applied
46 methods and their application. For a full description see
47 the open-access publication by <a
48 href="http://www.sciencedirect.com/science/article/pii/S1871101415300558">Knudsen
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 using a
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>), which
61 may vary over several orders of magnitude, are tested
62 with uniform probability across the logarithmic parameter
63 interval. The temporal parameter (<i>t</i><sub>degla</sub>)
64 and climate record threshold value
65 (δ<sup>18</sup>O<sub>threshold</sub>) are tested with
66 uniform probability across the linear parameter interval.
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 varied within
73 specified limits, they can be thought of as being orthogonal
74 axes spanning a coordinate system in four-dimensional space.
75 Each position in this model space is associated with a
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 duration
84 of e.g. the entire Quaternary period in a sample can be
85 computed. This <i>forward model</i> describes a possible
86 history of exhumation and TCN production in a sample volume
87 as it experiences the variable physical environment of the
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</h4>
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 comparable
98 intensity. Likewise, the erosion rate during the past
99 glacial periods is assumed to be comparable.</p>
100
101 <p>The model approach assumes that glacial periods were
102 characterized by 100% shielding and no exposure, which
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 shielding. The
107 production of TCNs takes place during the interglacials,
108 while erosion removes the land surface at different
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 record
113 crosses a threshold value. The provided climate records
114 are based on a benthic δ<sup>18</sup>O record,
115 smoothed by various degrees, implying that climate
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 entity
125 which sequentially explores the model parameter space in
126 order to obtain the closest match between the forward
127 model and the observational dataset of TCNs. During each
128 iteration the walker takes its current position in model
129 space, plugs the parameter value into the forward model,
130 and evaluates if the output result matches the
131 observational record better or worse than the output at
132 its previous position in model space. If the new results
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 space, a
139 burn-in phase of 1000 iterations is first used to make a
140 crude search of the entire model space. The burn-in
141 phase is followed by a similar but more detailed and
142 local search of the model space, based on the best-fit
143 model parameters from the burn-in phase. The weighted
144 least-squared misfit to observed TCN concentrations is
145 used to evaluate the likelyhood for the combinations of
146 model parameter values. The MCMC walker continues
147 exploring the model space until it is sufficiently
148 satisfied with the best model parameter estimate it has
149 found.
150 </p>
151
152 <p>
153 For a given observational data set more than one set of
154 model parameters may produce forward models which
155 sufficiently satisfy the MCMC walker. In this case the
156 solution is <i>non-unique</i>. Even worse, a single MCMC
157 walker may find an area in model space which seemingly
158 is in good correspondence with the observational data
159 set, but the walker is missing a much better set of
160 model parameters since they are located somewhere
161 entirely different in the model space. In order to
162 mitigate these issues, MCMC inversions are often
163 performed using several MCMC walkers. The starting
164 point of each MCMC walker is chosen at random, resulting
165 in unique walks through the model space. If a single
166 walker is caught in an area of non-ideal solutions,
167 chances are that the other walkers will find the area of
168 better model parameters.
169 </p>
170
171 <p>
172 The computational time depends on the number of MCMC
173 walkers. When casually trying out the calculator we
174 recommend using low numbers of MCMC walkers (1 to 2) in
175 order to obtain fast results and reduce load on the
176 server. When attempting to produce high-quality and
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 by
190 citing:</p>
191 <blockquote>
192 Knudsen, M.F., Egholm, D.L., Jacobsen, B.H., Larsen, N.K., Jansen,
193 J.D., Andersen, J.L., Linge, H.C., 2015.<br>
194 <b>A multi-nuclide approach to constrain landscape evolution and
195 past erosion rates in previously glaciated terrains.</b></br>
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"> @article{Knudsen2015,
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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