Lecture index: Introductory thoughts. / Markov Chain analysis of stratigraphic successions. /An example - peritidal carbonate sequences. / Exercise 13.

Chapt. 6 Data Through Time - Swan & Sandilands, 1995, Introduction to Geological Data Analysis, Blackwell Science. 213-264. Read 213-220 in detail. Just skim the rest.

Humans have a fascination with history. Through understanding history we come to understand process and guess at destiny. You may remember quotes along the lines of those who do not know history are doomed to repeat it. A basic question is as to the pattern or character of history. Does it repeat itself. This is captured in the discussion on time's arrow and time's cycle. The distinction between and the necessity for both is nicely explored by Stephen J. Gould in his book Time's Cycle, Time's Arrow, and is encapsulated in the qupte below.

"time's arrow and time's cycle is, if you will, a "great" dichotomy because each of its poles captures, by its essence, a theme so central to intellectual (and practical) life that Western people who hope to understand history must wrestle intimately with both - for time's arrow is the intelligibility of distinct and irreversible events, while time's cycle is the intelligibility of timeless order and lawlike structure. We must have both. " Stephen Gould, p. 15-16, 1987.

Mathematically one can represent time's arrow with a linear relationship with time as the x variable. Given a value of y there is only one unique value of t that can work. A sin or cosine function on the other hand can represent time's cycle. For a give value of y there are many possible times that it could correspond to.

Types of temporal patterns:

• episodic events vs. continuous history.
  • earthquakes occurrences vs. sea level history.
  • flash floods in dry arroyo vs. flow in the Missouri.
  • can you think of other types of episodic events?
• types of episodic event patterns (from Swan & Sandilands):
  • regular.
  • random.
  • clustered (beyond random clumps).
  • trend.
  • pattern.
  • in chaos lingo - strange attractors. Patterns of unpredictable behavior.
• types of continuous temporal patterns:
  • linear evolutions.
  • cyclic.
  • non-linear evolutions.
  • chaotic, long term practically unpredictable. (e.g. Verhultz equation).
  • random, short term unpredictable.

What are external forcing agents inducing cyclicity in the geologic record:

• daily cycles:
• tidal/lunar cycles:
  • tidal rythmites and bundles.
  • coral growth rings.
• seasonal cycles:
  • varves.
  • tree rings and dendrochronology.
  • atmospheric carbon dioxide levels.
• solar cycles (e.g. sunspot cycle).
• faulting: not necessarily with fixed units of time, but in terms of succession of events.
• Milankovich cycles: various parameters. Cycles of 30,000 to 400,000 years.

What geo-systems might have cyclic behavior driven by internal dynamics:

• pulsing stream flow.
• eruptions (filling and evacuating a chamber).
• definitely seen in organisms (circadian rythms).
• reservoirs with release thresholds and semi-constant feed a general scheme.
• super continent and overturn models cycles

A complicated approximation to the truth and how to capture it: Much of the above deals with simplified end-member types of behaviors. In reality one can expect mixtures. Cyclic sea level changes driven by glacial-eustatic coupling and orbital forcing are likely superimposed on long term tectonic changes, with a possible random noise component, due to other factors. An equation that might describe such behavior could look something like:

where y is sea level, and k, m, a, p and w are constants, and t represents time. The first two terms would represent a linear evolution if p=1 and a nonlinear powerlay relationship if t is not, and the last would represent a cyclic component with the constants a and w controlling the amplitude and wavelength respectively. By adding other terms we can introduce other cycle components and some very interesting behavior. The relative wavelengths and whether the cycles are in phase or out of phase are important considerations in the behavior. One could also build an exponential component with ease.

This lab will introduce you a bit into analysis of time series date for temporal patterns. As a perusal of the latter part of your reading suggests this can be a very complicated process and it can be. However, initial entry is not difficult. In addition to something called Markov chain analysis you will also explore some simple plots that can identify patterns in time-series event data.

Antarctic Vostok ice core data source:

Graph of T changes recorded in Vostok core as measured by hydrogen isotopes. Look at the pattern qualitatively. What do you notice about it? Here the pattern is strongly developed - obvious. Often it is not.

One place that Markov chain analysis has been used is in the analysis for cyclothems in sedimentary sequences. This can lead to the question - what is a cyclothem?

Cyclothems: A cyclothem is a model for a pattern of repeated facies migrations that leaves a distinctive reptititve pattern in the vertical stacking of lithologies (remember Walther's law). They are particularly well known from the Carboniferous, and also from the Cretaceous (Brenner & McHargue, 1988). They are developed for and can bear the name of a region and/or geologic period. The cause for the repetition can be internal factors (such as lobe switching in a deltaic environment) or external forcing agents such as eustatic changes. Better known examples include the Illinois cyclothem, which is thought to be a result of delta shifting, and the Pennsylvanian Kansas cylothem (see figure to left) which is attributed to glacial transgression-regression cycles on a shallow carbonate-shale shelf. Cyclothems related to faulting have also been explored. You might think of other geologic situations that might produce a repetitive vertical sequence or rocks.

Remember that humans are biased to seeing patterns even where they do not exist. This is why Markov chain and other types of even more rigorous statistical analysis are needed to establish that there is actually some sort of repetitive pattern and what the pattern is. Field work and a qualitative/intuitive assessment may lead one to develop a cyclothem model. Markov analysis can be used as a test of the hypothesis.

You can think of a sequence of lithologies as a 'chain' of letters and ask the question as to whether there is a pattern in this chain. Naturally, the longer your chain the more confidence you will have in the results of the analysis ("Please, sir, can I have some more data."). Markov Chain analysis is based on the simple question of whether a given lithology is independent or not on the lithology below. The greater the dependence the more likely that transition from one bed to another is part of a pattern of behavior. Technically this is known as a first-order Markov analysis. You can also ask to what degree a lithology is dependent on a sequence of lithologies below. If you look at the two underlying lithologies then this would be a second-order Markov analysis. The simple comparison is to what would be expected randomly (equivalent to independence). If 25% of our lithologies as represented in the chain, are limestone, then 25 percent of the time we would expect limestone to lie on top of any other lithology (including limestone) if stacking were random. Using a matrix we can look at all the observed transitions and see how they compare to a random stacking pattern. We will look only at first order Markov Chains.

Your reading does a very good job of describing the basis of first order Markov chain analysis. The key is constructing transition frequency and probability matrixes. You can model your analysis directly after the example in the book.

We are always interested in finding new places to use our skills and tools to profit. Can you think of other non-geologic applications where this type of analysis could provide insight.

Source of information and data: Journal of Sedimentary Research; November 1996; v. 66; no. 6; p. 1065-1078 © 1996 SEPM Society for Sedimentary Geology Facies successions in peritidal carbonate sequences, Bruce H. Wilkinson, Nathaniel W. Diedrich, and Carl N. Drummond

One might expect shallow carbonate platforms to be very sensitive to sea level changes. The table below is part of such a lithologic sequence, measured bed by bed, and the assignment of each lithology to one of four categories, and then the transition that is seen. The table below that is the Markov matrix analysis.

1)Analyzing for cyclothems in the Billefjorden basin of Spitsbergen.

You have (or will be given) a copy of a portion of a stratigraphic fence diagram from a Carboniferous basin in Spitsbergen (taken from Johannessen & Steel, 1992, Mid-Carboniferous extension and rift-infill sequences in the Billefjorden Trough, Svalbard). Authors claim cyclicity. These are sediments deposited next to a fault in a half graben. Cycles could be related to episodes of fault movement and sedimentary readjustment or to orbital or glacial/eustatic forcing or some type of combination. You will test for it. You should basically replicate the Markov Chain analysis as described in the first part of your reading for this diagram.

The lithologies you will be working with are a) limestone/dolomite, b) evaporites, c) fluvial sandstone and shallow marine sandstone and conglomerates (lumped together), d) and alluvial fanglomerates. You should thus have a 4 by 4 matrix and 16 possible transitions (similar to what is in your reading).

Your final product should include:

• a) a lithologic sequence derived at a chosen interval (something less than every 10 meters).
• b) a transition frequency matrix (easily done in Excel).
• c) a transition probability matrix.
• d) a diagram similar to Fig. B6.11 in your reading.
• e) the expected random transition frequency and count matrices.
• f) the results of a Chi-squared test with the null hypothesis that it is a random pattern.
• g) a 250 word or less description of what you think is going on sedimentologically.
• Remember - don't hesitate to come find me for help with this!!

2) Graphical analysis of a series of events - The Aso eruption case history:

Much of this is taken from Davis, 1986, Statistics and Data Analysis in Geology, 2nd edition. Here they consider a long history of eruptions for the Volcano Aso on Japan (Kyushu). The data is given below.

1229, 1239, 1240, 1265, 1269, 1270, 1272, 1273, 1274, 1281, 1286, 1305, 1324, 1331, 1335, 1340, 1346, 1369, 1375, 1376, 1377, 1387, 1388, 1434, 1438, 1473, 1485, 1505, 1506, 1522, 1533, 1542, 1558, 1562, 1563, 1564, 1576, 1582, 1583, 1584, 1587, 1598, 1611, 1612, 1613, 1620, 1631, 1637, 1649, 1668, 1675, 1683, 1691, 1708, 1709, 1765, 1772, 1780, 1804, 1806, 1814, 1815, 1826, 1827, 1828, 1829, 1830, 1854, 1872, 1874, 1884, 1894, 1897, 1906, 1916, 1920, 1927, 1928, 1929, 1931, 1932, 1933, 1934, 1935, 1938, 1949, 1950, 1951, 1953, 1954, 1955, 1956, 1957, 1958, 1962

Data in form easily loaded into Excel.

You may also use the Vostok ice core data if you want to explore that data set.

First, look at this data and see if you detect any pattern as you peruse it.

The following plots are simple to do in Excel and provide a first look for any pattern in the series.

A cumulative frequency plot: This plots the total number of events that occured at or before time t versus t. The slope between any two points on the plot gives the average number of events per unit time. A straight line would indicate constancy of occurrence. Distinct breaks in the plot would suggest a change in behavior throughout the history. Sorting your data will help make creating this plot fairly easy.

Histogram plot: Basically create a bar graph showing the number of events within time bins of constant interval. If a well developed trend or pattern is developed it may show up here. Remember that histograms are sensitive to the interval position and width.

Empirical survivor plot: In this plot x is the length of a time interval between events, and y is the proportion of event intervals in the record longer than x. The scatter plot best-fit line should have a negative slope since longer intervals are less frequent than smaller ones. This plot will have an exponential form if events occur randomly in time. To see departures more easily a log-log plot will turn an exponential form into a straight line and then departures from that line can be seen.

Serial correlation (first-order correlation) plot: this is plot of x as the interval before a given data point versus y the interval after a given data point. A plot with a lot of scatter and a higher concentration of points near the origin is typical of a random series of events. Think about what a plot would look like if intervals were spaced about every 10 then 20 year cycles. Two clusters symmetrically disposed should be evident.

Hand in these 4 plots (labeled) and describe your results in 250 words or less. Can you think of any problems with the data set that might obscure a pattern?

Note by the way that even if eruptions happen randomly or close to randomly through time, potentially they can still be predicted using precursors and a monitoring system.

References:

• Brenner, R. L. & McHargue, T. R., 1988, Integrative Stratigraphy - Concepts and Applications; Prentice Hall, 417 p.
• Davis, J. C., 1986, Statistics and Data Analysis in Geology, John Wily and Sons, 646 p.
• Gould, S. J., 1987, Time's Arrow - Time's Cycle; Harvard Press, 222 p.
• Herbert, T. D., Gee, J, & DiDonna, S., 1999, Precessional cycles in Upper Cretaceous pelagic sediments of the South Atlantic: Long term patterns from high-frequency climate variations.GSA Special Paper 332, p. 105-120. Nice example of orbital forcing.
• Miall, A. d., 1973, Markov chain analysis applied to an ancient alluvial plain succession. Sedimentology, v. 20, p. 347-364.

Copyright by Harmon D. Maher Jr.. This material may be used for non-profit educational purposes if proper attribution is given. Otherwise please contact Harmon D. Maher Jr.