a search for periodic and quasi-periodic patterns in...

APPROVED: James Roberts, Major Professor Yuri Rostovtsev, Committee Member Gary Glass, Committee Member Paolo Grigolini, Committee Member David Schultz, Chair of the Department of

Physics David Holdeman, Dean of the College of Arts

and Sciences Victor Prybutok, Vice Provost of the

Toulouse Graduate School

A SEARCH FOR PERIODIC AND QUASI-PERIODIC PATTERNS IN SELECT PROXY

DATA WITH A GOAL TO UNDERSTANDING TEMPERATURE VARIATION

James Otto, B.S., M.S.

Dissertation Prepared for the Degree of

DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS

May 2016

Otto, James. A Search for Periodic and Quasi-Periodic Patterns in Select Proxy Data

with a Goal to Understanding Temperature Variation. Doctor of Philosophy (Physics), May

2016, 164 pp., bibliograqphy, 192 titles.

In this work over 200 temperature proxy data sets have been analyzed to determine if

periodic and or quasi-periodic patterns exist in the data sets. References to the journal articles

where data are recorded are provided. Chapter 1 serves an introduction to the problem of

temperature determination in providing information on how various proxy data sources are

derived. Examples are given of the techniques followed in producing proxy data that predict

temperature for each method used. In chapter 2 temperature proxy data spanning the last 4000

years, from 2,000 BCE to 2,000 CE, are analyzed to determine if overarching patterns exist in

proxy data sets. An average of over 100 proxy data sets was used to produce Figure 4. An

overview of the data shows that several “peaks” can be identified. The data were then subjected

to analysis using a series of frequency modulated cosine waves. This analysis led to a function

that can be expressed by equation 3. The literature was examined to determine what

mathematical models had been published to fit the experimental proxy data for temperature. A

number of attempts have been made to fit data from limited data sets with some degree of

success. Some other papers have used a sinusoidal function to best fit the changes in the

temperature. After consideration of many published papers and reviewing long time streams of

proxy data that appeared to have sine wave patterns, a new model was proposed for trial. As the

patterns observed showed “almost” repeating sine cycles, a frequency modulated sine wave was

chosen to obtain a best fit function. Although other papers have used a sinusoidal function to

best fit the changes in the temperature, the “best fit” was limited. Thus, it was decided that a

frequency modulated sine wave may be a better model that would provide a more precise fit.

This proved to be the case and the more than 240 temperature proxy data sets were analyzed

using Equation 3. In chapter 3 the time span for the proxy data was extended to cover the period

of time 12,000 BCE to 2000 CE. The data were then tested by using the equation above to

search for periodic/quasi-periodic patterns. These results are summarized under select conditions

of time periods. In chapter 4 the interval of time is extended over 1,000,000 years of time to test

for long period “periodic” changes in global temperature. These results are provided for overall

analysis. The function f(x) as described above was used to test for periodic/quasi-periodic

changes in the data. Chapter 5 provides an analysis of temperature proxy data for an interval of

time of 3,000,000 years to establish how global temperature has varied during the last three

million years. Some long-term quasi-periodic patterns are identified. Chapter 6 provides a

summation of the model proposed for global temperature that can be expected if similar trends

continue over future years as have prevailed for the past few million years. Data sets that were

used in this work are tabulated in the appendices of this paper.

ii

Copyright 2016

by

James Otto

iii

ACKNOWLEDGEMENTS

My doctoral research work at the University of North Texas was made possible with the

support of many people.

I thank my major professor, Dr. James A. Roberts, for not only being my mentor, but also

a guiding force. His decades of experience, his enthusiasm, and his support in me helped me

remain focused on this project. I appreciate his dedication that has trained me to become an

independent researcher.

I also thank the contributions from my other committee members, including Dr. Paolo

Grigolini, Dr. Yuri Rostovtsev, and Dr. Gary Glass. It is with their constant discussions,

suggestions, and critiques that I have been able to stay focused on my research.

I would like to thank Ryan Lane, a physics candidate at the University of North Texas,

for his help with Mathematica coding and for being a good friend.

I would also like to thank the Department of Physics, its faculty and staff, for providing

me with an environment in which to learn and grow, and for providing me with the support that I

needed to get this work done.

The Toulouse Graduate School at the University of North Texas is especially thanked for

providing me the opportunity to earn this degree.

I would like to thank my mother and grandmother for their sacrifices and support in my

studies. I shall always remain indebted to their determination to see me finish my goal. I would

also like to thank my mother for her constant mailings of gift cards to see that this graduate

student ate well.

iv

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ........................................................................................................... iii

CHAPTER 1: INTRODUCTION ....................................................................................................1

1.1 History......................................................................................................................1

1.2 Proxy Data ...............................................................................................................4

1.2.1 Tree Ring Spacing........................................................................................6

1.2.2 Ocean Sediment ...........................................................................................8

1.2.3 Ice Core Samples .......................................................................................10

1.2.4 Pollen .........................................................................................................12

1.2.5 Chironomids ...............................................................................................13

1.2.6 Coral Reefs.................................................................................................13

1.3 Time Scales ............................................................................................................13 CHAPTER 2: ANALYSIS OF PROXY DATA FOR 4000 YEARS OF TEMPERATURE RECONSTRUCTION....................................................................................................................15

2.1 Method ...................................................................................................................16

2.2 Temperature Reconstruction without Using the Scandinavian Data Sets .............30

2.3 Conclusion .............................................................................................................41 CHAPTER 3: ANALYSIS OF PROXY DATA SETS OVER 14,000 YEARS OF TEMPERATURE RECONSTRUCTION .....................................................................................43

3.1 Abstract ..................................................................................................................43

3.2 Introduction ............................................................................................................43

3.3 Methods/Procedures ...............................................................................................43

3.4 Results ....................................................................................................................57

3.5 Temperature Reconstruction without Using the Scandinavian Data Sets .............57

3.6 Conclusion/Discussion ...........................................................................................67

CHAPTER 4: EVALUATION AN EXTENDED PERIOD OF TIME FOR 1,000,000 YEARS OF TEMPERATURE CHANGE RECONSTRUCTION ..............................................................70

4.1 Abstract ..................................................................................................................70

4.2 Introduction ............................................................................................................70


v

4.3 Results ....................................................................................................................77

4.4 Conclusion/Discussion ...........................................................................................81 CHAPTER 5: EVALUATION OF AN EXTENDED PERIOD OF TIME OF 3,000,000 YEARS OF CLIMATE TEMPERATURE RECONSTRUCTION ............................................................83

5.1 Abstract ..................................................................................................................83

5.2 Introduction ............................................................................................................83


5.4 Results ....................................................................................................................90

5.5 Conclusion/Discussion ...........................................................................................94

CHAPTER 6: SUMMARY, CONCLUSIONS AND PREDICTIONS FOR FUTURE WORK ........................................................................................................................................................96

APPENDICES .............................................................................................................................101 BIBLIOGRAPHY ........................................................................................................................149

CHAPTER 1

INTRODUCTION

“Although there is considerable uncertainty with respect to the rate and magnitude of any

future warming which may occur as a result of human activities, one aspect of this question is

not in dispute: any human-induced change in climate will be superimposed on a background of

natural climate variations. Hence, in order to be able to predict future climate changes, it is

necessary to have an understanding of how and why climates have varied in the past” (Daux, et

al., 2005).

1.1 History

Global warming continues to be of prime concern to those who want to preserve our

present environment. For the past 160 years, mankind has been keeping track of the temperature

accurately, and during that time, an increase in the overall average temperature has been

observed. This record of the temperature gives us a good short-term indication of climate

change, but it is minute on the geologic timescale, and needs to be combined with longer term

data to establish long-term patterns in temperature. To understand the temperature changes

beyond these short periods, one needs to look at proxy data to fill in the missing data from past

time intervals. The validity of proxy data can then be ascertained by comparing proxy data that

overlaps with data obtained using direct measurements.

There are many ways of looking at the past in terms of temperature change, mostly by

looking at “temperature gauges” that will allow us to determine temperature trends in the past

centuries. These indirect indicators of temperature have to be relied upon to obtain long time

temperature trends. These long-term proxy data will give us a better understanding of the

1

temperature beyond what our modern records show. In addition, by comparing trends in proxy

data from different indicators, i.e. tree ring growth patterns, ice core deposits, pack ice records,

etc., further verification in trends can be established for temperature.

There are a number of papers in the literature on the topic of global temperature

reconstructions. Some examples of these are given here. There is a paper by Dr. Craig Loehle in

which he made a 2000 year global temperature reconstruction based on non-tree ring proxies

(Loehle, 2007). In his paper, he used 18 temperature sets to find the temperature deviation. This

paper did not go into finding periods or best fit models for the data but was limited to just

combining the temperatures into one set. In this dissertation over 240 temperature

reconstructions were combined into four different sets that spanned four different time periods.

Another paper by Loehle and Siegfried Fred Singer used nine temperature

reconstructions to create a best fit model and came up with a 1470 year cycle (Loehle & Singer,

2010). In that work a simple sinusoidal wave function was employed to fit the data.

In a third paper, Craig Loehle and Nicola Scafetta attempt to show a 20 year cycle and 60

year cycle, covering the time period of 1850 to 2010 (Loehle & Scafetta, 2011). This limited

time frame is not a long enough time frame to obtain a reasonable representation of a meaningful

periodic occurrence. In that limited work they used two sine waves (assuming 60 year and 20

year periods) and fit the data to a best fit linear line so that the best fit line indicates that the

temperature rises over time. Also, the interval of time is for a time period that is during the

industrial revolution, a very controversial period of time. In the paper they ignore any changes in

temperature that might be manmade, and instead attempt to show a natural fluctuation during the

period of time chosen for study.

2

It is well known that the Earth goes through periodic cycles as the axial tilt of the Earth’s

spin axis relative to the orbital plane fluctuates, the change in its eccentricity of its orbit as it

circles the Sun, as it rotates on its axis, and as it interacts with the gravitational pull of the other

planets. The interaction of these bodies with the Earth and with each other, affects the amount of

solar radiation arriving at the Earth and this in turn will modify the climate of the Earth. If the

Earth does experience periodic and quasiperiodic interactions that effect the temperature, then

the evidence for these periodic changes should be evident in the temperature changes that are

seen in the proxy data of past temperature changes. To test for this effect and to determine any

quasi-periodic patterns in the temperature over time, a study was made of over 200 proxy

temperature data sets to test for such periodic and quasiperiodic patterns that are found in the

proxy data from around the world. An overview of a number of long term proxy temperature

data sets indicated patterns that tended toward “periodic”. It was assumed from these

preliminary observations that a standard frequency modulation techniques could be used to

model these periodic patterns. Each one of the “periodic” events associated with planetary

motion that changed the distance of the Earth from the Sun or modified the angle of incident of

solar radiation at the Earth would manifest itself in the temperature at the site of the Earth.

In this work a proposal is made to add to the temperature modeling problem in four

different ways. In the first aspect, a model is constructed using a frequency modulated

sinusoidal function instead of a simple linear function or a standard sinusoidal function.

In the second aspect of this work, a larger time frame than other papers have employed

will be explored. Most authors produced papers that have been limited to short term intervals,

consisting of only the past few hundred to a few thousand years, while in this paper, the time

interval has been extended to include short term and long term time periods of 4 kYrs, 14 kYrs, 1

3

MYrs, and 3 MYrs to test this model. This gives a larger range of times to determine what

periods occur in overarching data sets for temperature.

A third aspect of this paper is to determine the natural periodic fluctuations that may

occur in the temperature independent of any human contributions. i. e., prior to the Industrial

Revolution. Thus, temperatures after 1700 CE are not included in the data used to create a best

fit model. Any temperatures that have increased with the introduction of the industrial

revolution are not included in the natural temperature changes. Some other papers include the

temperature changes throughout the industrial revolution in their modeling, which makes the

assumption that global warming is a natural function instead of manmade.

The fourth aspect of this paper is to extend the analysis of proxy temperature data to

encompass more than 240 different temperature reconstructions in determining the average

temperature reconstruction. This procedure will greatly improve the analysis in that the more

sources that are included will provide a more meaningful global representation of the

“background” temperature.

1.2 Proxy Data

In order to determine past temperature changes beyond the past two centuries, data

designated as proxy data are used. The proxy data will provide a record of the trends in

temperatures of centuries past, but will not provide a direct measurement of the temperatures

themselves. From the proxy data it can be determined if the temperatures was warming or

cooling, and by comparing these data with other proxy data and recent recorded temperatures, a

reasonable assessment can be made of the past climate. In order to have a better understanding of

4

global temperature change, a broad spectrum of data from around the world needs to be

collected.

The temperature reconstructions for multiple regions that was used is this dissertation has

been collected from the National Oceanic and Atmospheric Administration (NOAA)

Paleoclimatology Data website (National Oceanic and Atmospheric Administration, n.d.). The

World Data Center for Paleoclimatology (National Oceanic and Atmospheric Administration,

2008) host multiple types of data from solar, geophysical, environmental, and paleoclimate data.

The goal of the World Data Center for Paleoclimatology is to document each data set so that it

can be archived for future study and to make this paleoclimate data and information freely

available without restrictions.

The NOAA relies on the contributions from scientists in the scientific community to

archive the results of their published research. What this allows for, is that the data is quality

controlled, which means that it has been peer reviewed by scientists that are experts in

paleoclimate data and research. All that is asked of the people that use the data is to cite the

original publication to give credit where credit is due.

There are many ways of looking at the past temperature changes. These include tree ring

spacing, δ18

O in ice core samples, ocean sediment, pollen, and chironomids. Each of these

proxies provides us with information that connects these events with changes in the environment.

Although they are not direct indicators of temperature, they fluctuate based on the temperatures

present in the environment, which in turn manifests itself as temperature changes over the Earth.

5

1.2.1 Tree Ring Spacing

One way of looking at past temperature changes is by observing changes in tree ring

spacing over a period of time. At the beginning of the last century, A. E. Douglass was able to

show that the tree ring growth in trees in semi-arid sites correlated with variations in the climate

(Fritts, 1966). It was concluded that one way of looking at past temperature changes could be by

observing tree ring spacing as they change over time. The method has been shown to be accurate

enough to date structures in Mesa Verde, Arizona, that have no historical record of the time of

construction.

One of the simplest observations of tree ring growth is that the age of a tree can be

determined by counting the rings of the tree, and that every year, the tree produces a new tree

ring. In this way, one can tell how old a tree is by counting rings. The number of rings

determines the age of the tree. The ring spacing and shape provide information on the amount of

rainfall, temperature and general growing conditions to which the tree was subjected over its

lifetime. Tree ring spacing is not uniform, and that some years leave behind wider rings, while

others leave behind very narrow rings. Scientists use the information about tree rings to get

information about the past growing conditions. The study of tree rings to obtain information for

the past climate change is called dendroclimatology (Fritts, 1976). The use of tree rings is very

precise when used for dating, since tree rings can be matched up with other trees in the same

general area. By matching the wide rings and narrow rings together, the timeline of that area can

be extended in a process known as cross-dating (Fritts, 1976), and assign years to each ring.

When scientists use this method and match a tree ring sample to an old sample of tree rings, the

outermost rings on the old sample indicate when the tree died (Fritts, 1976). This is precise

enough that archeologist can use this method to date Native American remains such as the

6

Pueblos by looking at the tree rings on the wood used in the construction, and matching them to

known tree rings to see when the wood was cut down and used for construction. The same

method applies when historians try to date an event, such as a natural disaster (fire, flood, etc.).

Historians can examine a tree that was destroyed during the event, and determine when an event

occurred by matching the tree rings. Tree rings provide such precise dating because the tree ring

growth is affected by changes in temperature and the yearly changes of favorable conditions such

as wet and/or warm climate and unfavorable such as dry and/or cold climate (Fritts, 1976). This

information is recorded by the sequence of wide and narrow widths of the tree ring growth

patterns. During the year when the growing conditions are favorable, the tree ring spacing is

wider than during less favorable times.

Thus, tree ring chronologies are a good way of developing long histories of temperature

variation for studying temperature change (Cook, 1985). The thickness of the tree ring can tell

you about the temperature and moisture changes related to the climate. When the climate is

warming, the growth of the trees increases, which results in tree rings that will be wider. When

the climate is cooling, there is a decrease in the growth of trees, and thus their tree rings will be

more closely spaced. So by observing the spacing of tree rings, one can determine the warmer

and colder periods in history. To reconstruct the temperature in the past, one must take the

following steps. First, we need to compare modern temperature records with the width of tree

rings corresponding to the same period of time. Then establish a statistical equation for the

relationship between the two sets of data. Third, we can then substitute the tree ring widths that

have been dated in the equation to get a better indication of what the temperature was during the

previous years for a particular tree. (Fritts, 1976).

7

Indirect data that is used to make temperature predictions is considered proxy data since

it doesn’t tell us the actual temperature directly during that time, but provides us an estimate of

what the temperature was during a specific time and enables changes in temperature to be

determined.

Not only can tree ring growth patterns be used from recent trees, but a fortuitous event

occurred in the state of Colorado, USA, in which trees were extinguished about 35 million years

ago by volcanic ash and then petrified at a later date. The tree ring growth pattern was preserved

in the petrification process and remains as a record of temperature to be studied today. The

pattern of growth indicates larger tree ring growth on average around 1.4mm compaired to

similar trees of today with mean ring growth of .84 mm to .96 mm. This indicates that the Earth

enjoyed a warmer climate than experienced today during the growing time of the trees (Gregory,

1992).

1.2.2 Ocean Sediment

Another method to determine the climate is by studying the sediment deposits on the

ocean floor. Since sediments pile on top of the preceding layer, one can analyze the layers of

sediment from the ocean floor and indicate the history of temperature change. In the sediment,

we find the shells of foraminifera, which can be used as a proxy of seawater temperature

(Chamberlin & Dickey, 2008). It was discovered by Cesare Emiliani that the measurements of

18O that was found in the shells could tell us about the ocean temperature (Emiliani, 1955). We

can even look at oxygen isotopes in coral to get an idea of the temperature in the ocean

(Guilderson, et al., 2001) (Felis & Rimbu, 2010). This is done through the use of a simple model

based on thermodynamics. For a given temperature, the kinetic energy will cause gases or liquids

8

to evaporate inversely proportional to their mass. Thus, for select isotopes of a gas the ones with

the lower masses will evaporate before the more massive ones. This change in the ratio of the

two isotopes of oxygen can be utilized to make temperature estimates. The variations of the

18O/

16O (δ

18O) in the water and ice cores are an important indicator of the temperature (Luz, et

al., 2009).

When studying the ocean sediment deposits, scientists drill into the ocean floor, around

200 – 500 meters deep, and create a core of sediment deposits meters in length. In the core, one

can study the organisms in the sediment and analyze the oxygen ratios that are trapped in the

sediment, to get an estimate of what the temperature was in the past. The seashells maintain their

integrity concerning the oxygen ratios embedded in them at the time that they ceased to process

oxygen. It is even possible to make measurements of the oxygen isotope ratio from silica from

the sediments of high altitude lakes (Barker, et al., 2001).

It is well known that an oxygen molecule has 8 protons and 8 neutrons, which is 16

O. A

small amount of oxygen is found to be 18

O which is an isotope of oxygen with 8 protons and 10

neutrons. This isotope is more massive than its 16

O counterpart, and won’t evaporate as readily

as the less massive 16

O. The vapor pressure of the water molecule H216

O is higher than the vapor

pressure of H218

O. So when water evaporates, a heavier concentration of the smaller mass water

molecule with 16

O will evaporate, leaving the water in the oceans more enriched with water

molecules containing 18

O. So, during a warm period, when there is a lot of evaporation going

on, the amount of 18

O water molecules will increase compared to the 16

O water molecules, since

the 16

O water molecules are more readily evaporating than the 18

O water molecules. Thus, this

ratio of 18

O water molecules to 16

O water molecules can give us a reasonably good estimate for

past global temperatures. Even though absolute temperatures may not be determined from these

9

ratios, relative changes can be ascertained from measurements of the ratio of 18

O to 16

O in water

molecules.

1.2.3 Ice Core Samples

Similarly, the ratio of oxygen isotopes can be used to determine the past temperature by

looking at the oxygen isotopes in the water ice core samples from glaciers near the geographic

poles of the Earth. Using the same process as above, one can measure the 16

O to 18

O ratio in the

ice and determine the temperature at the time of formation of a specific ice deposit. As was

mentioned before, water molecules with 16

O atoms evaporate more readily than their more

massive counterpart, but at the same time, water vapor molecules containing the more massive

18O condense more readily (Riebeek, 2005). The water that evaporated near the equator,

containing 16

O and 18

O, moves toward the poles. During the cycle of pole-ward transportation of

the water vapor, the different isotopes continue to change the fraction of 18

O water molecules to

16O water molecules, with the heavier

18O water vapor being reduced by falling as precipitation.

During condensation, because the water molecule H218

O has a lower vapor pressure, it can pass

more readily into the liquid state than can the water made up of the lighter oxygen isotope

(Dansgaard, 1961). Because the condensation of the water vapor into liquid is a result of

cooling, the larger the decrease in temperature, the smaller the heavy isotopes of water

concentration will be. This leaves the lighter 16

O in the clouds, which slowly over time have

been depleted of its H218

O water vapor. Since the water vapor with the heavier 18

O molecule

condenses more readily than the 16

O water molecules, the precipitation closer to the equator, due

to the higher average temperature, will be richer in the water molecules containing 18

O while

10

closer to the poles, the remaining water with 16

O will precipitate, making the snow at the polar

region have a higher concentration of the less massive oxygen.

During colder times, the lower temperature that would normally be found at the poles

extends closer to the equator, so the heavier water molecules containing the 18

O rain out of the

atmosphere at even lower latitudes then under normal conditions. As the air cools, either by

rising into the atmosphere or moving towards the poles, the moisture from the water vapor in the

air begins to condense and fall as precipitation. Initially, the water will contain the 18

O water

molecules and the 16

O water molecules in a given ratio. In due course, the the water vapor in the

air will be depleted of the heavier oxygen isotope 18

O water molecules to change the ratio of the

two isotopic concentrations (Riebeek, 2005). This means that the water around the equator is

going to have a higher concentration of 18

O. The lighter 16

O isotope water vapor moves towards

the poles, and eventually by the time the water vapor reaches the poles, the majority of the water

vapor is 16

O and eventually condenses into water and falls, where it lands on ice sheets to form

layers (Riebeek, 2005). Overall the water in the ocean develops a higher concentration of

“heavy” oxygen near the equator as compared to the universal standard, and the ice near the

poles develop a higher concentration of the lighter oxygen (Riebeek, 2005). This selection

process is what we expect to see during Ice ages.

During warmer times, the water is rich in light oxygen, where ice sheet melts have

returned freshwater into the ocean. This is opposite to the high concentration of the 18

O water

molecules observed during cooler times. The reason for this is because during warmer times, the

ice at the poles melt, and this melted water then flow into the ocean. This freshwater contains

more of the lighter 16

O isotope water than 18

O water molecules (Riebeek, 2005). Now the

question to ask is “How one can use the isotope ratio that arises from the ice core samples and

11

from sediment deposits to determine relative temperature changes?” A technique has been

developed by Craig to determine this.

The change in ratio for the different oxygen isotopes, δ18

O, is a ratio of the oxygen

isotopes compared to the Standard Mean Ocean Water (SMOW) standard (Craig, 1961). By

knowing the difference between the two ratios, the change in temperature can be determined.

Equation 1 can be used to express this ratio.

Equation 1

Stable isotope data can serve as a universal proxy indicator of the past climate variables

that are recorded in many formats, such as in sediment and ice cores, and can be directly

compared, not only with each other, but with other proxy data (Daley, et al., 2011).

1.2.4 Pollen

Another way to study the past climates is to observe pollen that has accumulated over the

years in sediment. Pollen comes from a wide range of plants, each with their own unique shape

and size, which makes it unique to a particular plant. In addition,, they have an outer wall made

up of a substance known as Sporopollenin, which gives the outer wall a strong exterior (Osborne,

2013).

So, with a hard exterior, these pollen samples survive being blown around and washed

down rivers to get to lakes or the ocean. Once there, the pollen settles onto the settlement and

over time the new sediment covers over the old sediment, giving us layers of sediment with

12

pollen in it. Scientists can then look at the shape of the pollen, and determine the plants that were

alive at that particular period of time. One then needs to date the sediment depth to a particular

time period, so that what pollen spores were present then can be determined. Using the

knowledge of what plants grow in a particular temperature, the temperature at that time can be

determined.

1.2.5 Chironomids

In northern latitude areas, another method of determining the temperature in the past can

be used. Chironomids, also known as nonbiting midges, is a type of fly that spends a large

portion of its life cycle in the larvae stage. Fossilized chironomids can be found throughout

sediment deposits, and can be a good indicator of the temperature in that area (Larocque, et al.,

2001).

1.2.6 Coral Reefs

Coral reefs are essentially living systems with dynamical growth patterns. Each year

layers of coral deposits form in a similar way to adding rings in growing trees. A cross section

of these patterns in coral deposits can be studied with a correlation between the growth pattern

and the environmental growing condition, i. e., mainly temperature. Thus, a study of coral reefs

enables proxy temperature data to be determined.

1.3 Time Scales

In this paper, the units of BCE (Before Common Era) and CE (Common Era) are used

instead of the more popular time unit of BP (Before Present) in order to allow the average person

reading this paper to better understand the timeframes discussed. Using Before Present (BP)

signifies the year before 1950 in which the temperature corresponds. Negative years would then

13

indicate years after 1950, such that 30 BP would be 1920 CE while -45 BP would be 1995 CE.

Majority of the data samples collected used the BP system, with only a few papers having their

data corresponding to the years before 2000, which is much less common. All data sets were

converted over to the BCE/CE nomenclature, so that the data sets were using the same

categorization.

14

CHAPTER 2

ANALYSIS OF PROXY DATA FOR 4000 YEARS OF TEMPERATURE

RECONSTRUCTION

In order to get a better understanding of the short term temperature change, on the order

of a few thousand years, a global temperature reconstruction for this time period has to be made.

All proxy data temperate reconstructions that were available in the literature that covered 4000

years or more were tabulated. Those temperature reconstructions that had at least 10 data points

within the timeframe of 2000 BCE to 2000 CE were chosen for study.

The data sets that fit these criteria and represented a global distribution were selected for

study. There were 109 sources that fit the criteria. These data sets were used for this

temperature reconstruction and are listed Appendix A. As one can see in Figure 1 the different

temperature reconstruction sources represent a global distribution that takes into account a

reconstruction on each continent. It is important to note that there is a cluster of data points in

the Scandinavia region that might skew the data, so another reconstruction of the temperature

was made without data from those sets included.

The temperature reconstructions for multiple regions that was used is this dissertation has

been collected from the National Oceanic and Atmospheric Administration (NOAA)

Paleoclimatology Data website. (National Oceanic and Atmospheric Administration, n.d.)

The data were then analyzed using Mathematica wherein the average of the data sets was

calculated. The average is then subtracted from the data set to set all the data sets around zero.

Before this, some data sets were centered around 24°C while others are centered around 3°C.

Comparing these data sets does not give a full understanding of the temperature changes between

them. So, by subtracting the average of the temperature for the data set, the different data sets

15

are referenced to the same level, namely around the 0°C mark, with temperatures larger than 0°C

being above average and temperatures smaller than the average as below average.

Figure 1: A global distribution of temperature reconstruction locations used in analyzing a 4,000

year span. Figure created in Mathematica 10 using the latitude and longitude coordinates for

each temperature site. Some sites overlap each other.

2.1 Method

Once this data is centered on 0°C, linear interpolation was used to fill in the interval

between some data points. This was done using Equation 2.

Equation 2

16

By using this equation, where and are two data points in the temperature

reconstruction, the interval between data points can be reconstructed. Although the temperature

may not change linearly between points, this procedure was chosen to “smooth” some of the data

sets. Once each of the data sets was interpolated, they were averaged together to get a general

trend for temperature change.

Some data sets were not considered in the series if they failed to meet a few simple

requirements. A data set had to have at least 10 data points in the time frame to be considered.

Fewer data points than this may not provide a reasonable indication of climate change in a

particular interval or location.

Figure 2: A graph of the number of sources from Table A.1 that was used for each decade in the

17

temperature reconstruction for the past 4000 years. The number of data sources becomes sparser

as proxy data for very ancient times is collected.

Using 109 proxy data sources, for the past 4000 years extending from 2000 BCE to 2000

CE some proxy data sets were analyzed. As can be seen in the following figure, when 109

sources of data are combined, the following results are obtained. From the graph, it can be seen

that the patterns show “dips” and “peaks” over time. These changes correspond to events in

recorded history. For example, a global cooling occurred around 1600 CE, giving rise to the

little ice age (LIA) which occurred throughout Europe. The Thames River near London froze

over for the only time in recorded history. It can also be seen that there was a rise in temperature

peaking around 1000 CE. This period is known as the Medieval Warm Period (MWP) which

occurred throughout the northern hemisphere during which the Vikings started a colony in

Greenland.

18

Figure 3: The graph of the average temperature reconstruction (in black) from the sources in

19

Table A.1. In blue is the error bars associated with each decade. The largest error in this

reconstruction is ± 0.207 °C.

This graph shows that the data that were collected and processed correlate with some

events in recorded history, and is, therefore, a good indicator of the reliability of the longer term

temperature data.

Figure 4: An average global temperature reconstruction of the past 4,000 years. Data adapted

from data sets in

20

Table A.1 in Appendix A.

Patterns in this graph, raised the question of whether there was periodic and or quasi

periodic fluctuations in the temperature, but at the same time, it is known that there has been an

increase in the global temperature since the industrial revolution. A model of the temperature

reconstruction without using the temperatures that correspond to the Industrial Revolution was

constructed. The precise beginning of the industrial revolution varies from source to source,

with some having it begin around 1750 CE and others having it begin as late at about 1800 CE.

To eliminate this confusion in the data that were collected, all data after the year 1700 CE was

not included in the first analysis. This way the only data that is used is data from the preindustrial

revolution, and the analysis will give a good idea of the way the temperature fluctuates naturally.

The choice of modeling the temperature change that occurs naturally, or at least with

minimal human interference, is that it gives us a base line. This base line of naturally occurring

temperature change is what human made temperature change will be super imposed upon, and

thus very important to understand.

To model the changes in the temperature, and to create a best fit function, it is necessary

to take into consideration the fact that the complex motion of the Earth can affect the temperature

changes at the site of the Earth. A model proposed by Milankovitch ( ilankovi , 1941)

describes how the Earth’s movement within the gravitational field provided by the Sun and the

other planets besides Earth can affect the climate of the Earth through perturbation of the orbit of

the Earth about the Sun. Such changes as the eccentricity of the Earth’s orbit due to orbital

perturbation, axial tilt of the Earth’s spin axis relative to the orbital plane, perihelion and

aphelion and apsidal precession of the Earth’s orbit all need to be considered in understanding

the complex patterns in the data sets.

21

The Earth has a very small elliptical orbit, with a value of 0.0174. But the value of the

eccentricity can fluctuate from 0.005, which is very close to circular, all the way to 0.06, because

of the gravitational forces exerted on Earth by Jupiter alone (Caissie, 2003). The eccentricity of

the Earth’s orbit has 2 different periods, one close to 100,000 years and another of near 400,000

years.

The average axial tilt of the earth is at 23.4 degrees. This tilt fluctuates between 22.1

degrees and 24.5 degrees over the course of a 41,000 year cycle.

The precession of the Earth has a period of about 26,000 years which indicates a potential

“periodic” modulation of the solar energy received by different latitudes of the Earth that

changes over a period near 26,000 years.

Since the Earth experiences multiple periodic fluctuations that can affect its climate,

either in a significant or a minor way, it seemed reasonable to choose a “sinusoidal” function to

analyze the data sets for periodic/quasi-periodic changes. The changing gravitational force on the

Earth’s orbit eccentricity as the planets move in their complex pattern will produce a modulation

of the periodicity of the Earth’s motion about the Sun. Thus, a more precise model requires a

modulated sinusoidal pattern. A sinusoidal function with frequency modulation was chosen to

model the temperature changes. The equation used is given by Equation 3.

Equation 3

The number of sine components was limited to N=8 frequency modulated sinusoidal

functions to best fit the climate reconstruction. In Equation 3, an and cn are amplitudes for the

fluctuations, bn and dn are the phases of each time event. The variable en is the modulation levels.

22

Equation 3 was adopted as a metric to test the data sets for periodic and quasi-periodic

fluctuations for periods dictated by the reference data set used to generate this basic equation.

The best fit curve was produced using Equation 3. The parameters obtained are listed in

Table A.2 which can be found in Appendix A. This best fit was limited to the time interval from

2000 BCE to 1700 CE. If the temperature is explored over the extended range 2000 BCE to

1700 CE, Figure 5 emerges. When the best fit curve in Figure 6 is compared with the best fit

curve in Figure 7, significant changes are found in the model that was created with data before

the industrial revolution and that observed when data during the industrial revolution are

considered.

Figure 5: An average global temperature reconstruction of the past 3,700 years from 2000 BCE

to 1700 CE. Data adapted from data sets in

23


Figure 6: The average temperature reconstruction for 3,700 years with best fit function. The

smooth curve is produced using Equation 3. Data used to plot this graph can be found in Table A.1

24

in Appendix A. Coefficients to Equation 3 that were calculated for the best fit line can be found in


Figure 7: The average temperature reconstruction for 4000 years with the best fit function using

data from 2000 BCE to 1700 CE. Data adapted from

25


The coefficients to Equation 3 that created the function you see in Figure 6 and Figure 7

can be found on Table A.2 in Appendix A. Looking at the table, also included in the table are

the period (in years) and the amplitude (in °C) that corresponds with each independent function.

Most of these functions are very minor, with amplitudes as small as 0.002 °C and can be ignored

as too minor to matter in understanding the big picture of temperature change.

Other functions were discarded as playing an important role in the modeling of the

temperature changes due to having too small of a period. Since the temperature reconstruction

used a temperature point every 10 years to make the overall temperature reconstruction that was

used to get the best fit line, periods that were too small were deemed insignificant to the overall

picture.

With this in mind, the functions that correspond to N=4 and N=5 were the only two in

which the function played a major role in best fitting the temperature changes from the sources in

26

Table A.1. The function that corresponded to N=4, had a period of 4450 years with an

amplitude of 5.2°C. The other function N=5 had a period of 4880 years and an amplitude of

5.1°C. These two functions set the overall trend of the best fit temperature line as can be seen in

Figure 8. Two smaller functions, N=1 and N=2, played a much more minor role. With the

function N=1, having a period of 714 years and an amplitude of 0.07°C, and N=2, with a period

of 235 years and an amplitude of 0.02°C, the amplitudes of these functions cause them only to

slightly influence the best fit line.


data from 2000 BCE to 1700 CE. The best fit function to produce the smooth curve in the figure

used only the N=4 and N=5 components. Data adapted from Table A.1 in Appendix A. The

27

coefficients used are in Table A.2 in Appendix A.

As can be seen in Figure 8, the two main functions chosen provide a reasonable

indication of the temperature fluctuations. Another method to smooth the plot is to consider only

the dominant periods and recalculate the best fit curve. In this case, every 50 years was averaged

and centered in that time span. The standard error of the mean was then calculated by taking the

standard deviation of the mean and dividing it by the square root of n (Bevington & Robinson,

2003). The new data were then processed using Mathematica and a new set of functions created

to best fit this smoothed data set. This can be seen in Figure 9, with the smoothed data (in blue)

and the best fit line in red.

In this run, the smoothed data sets allowed for larger periods to be calculated without all

the smaller “noise” that was calculated in the original best fit set. In this smoothed data set,

periods of 2120, 3120, 10,300, and 185,000 years were calculated. The large period functions,

such as the 185,000 and 10,300 years functions, were periodic but not sinusoidal, with multiple

smaller fluctuations within these periods. These larger period functions, 10,300 years and

185,000 years, are much larger than the time frame that is being considered, and are assumed to

be an artifact generated by the method of analyzing periods, until they are verified by spans of

time longer then a scanned period. Mathematica is simply a software package that is just

analyzing the data; not providing the meaning behind the numbers it has been given, and so it is

up to the observer to analyze the results that Mathematica has developed to provide meaning to

the results.

28

Figure 9: The smoothed temperature reconstruction for 3700 years with the best fit function

using data from 2000 BCE to 1700 CE. The error bars are the standard error of the mean. Data

adapted from

29

Table A.1 in Appendix A. The coefficients used are in Table A.3 in Appendix A.

2.2 Temperature Reconstruction without Using the Scandinavian Data Sets

The question arises that, being that there are many temperature sources from the

Scandinavia region, will this skew the data so the model will favor that region more. In this

section the Scandinavia data sets of temperature sources from that region were removed and the

limited data reevaluated. The following analysis includes only the global data set without the

Scandinavian data sets.

Specifically, the data sets from Finland, Sweden, and Norway were not included in the

following analysis. The 63 sets of sources from which the data sets were obtained can be found

in

30

Table B.1 in Appendix B.

Figure 10: A map of the global distribution of temperature reconstruction locations used in

analyzing a 4,000 year span without using the data sources in the Scandinavian region. Figure

created in Mathematica 10 using the latitude and longitude coordinates for each temperature site.

Some sites overlap each other.

31

Figure 11: A graph of the number of sources from Table B.1 that was used for each decade in the

32

temperature reconstruction for the past 4000 years. The number of data sources becomes sparser

as proxy data for very ancient times is collected.

A new combination of data that doesn’t include the large set of data from the Scandinavia

region was considered, the data was analyzed again in order to obtain a temperature

reconstruction without the Scandinavian data sets. These results are plotted in Figure 13.


33

Table B.1 in Appendix B. In blue is the error bars associated with each decade. The largest

error in this reconstruction is ± 0.301 °C.

Figure 13: Temperature Reconstruction without using data sources from the Scandinavian

countries. Data used to plot the graph are adapted from

34

Table B.1 in Appendix B

The changes are more pronounced, but otherwise the trends are very similar to the

temperature reconstruction which included the Scandinavia sources. Using Equation 3, a best fit

curve using only the data from 2000 BCE to 1700 CE was obtained and plotted in Figure 14.

Figure 14: A plot of the raw data for the global proxy data sets and the profile predicted by using

Equation 3. The data sources used in the reconstruction are listed in

35

Table B.1 in Appendix B.

When the best fit function is extended to the year 2000 CE, it can be seen that there is a

rise in the temperature as predicted by the model that is 0.2 degrees lower than what is observed.

This can be seen in Figure 15.

The continuing decrease in temperature for the past 200 years is not predicted by the

model when the Scandinavian data are not used. Further analysis is indicated to understand why

the local temperatures of this region predict lower temperatures than are experienced today.

Figure 15: A reconstruction of temperature over the time period 2000 BCE to 2000 CE. The

black curve is the raw data. The smooth red curve is the temperature trend predicted by Equation

3. The data sets used in the reconstruction are listed in Table B.1 in Appendix B. The

36

coefficients for the best fit line are in Table B.2 in Appendix B.

The coefficients to Equation 3 that created the function you see in Figure 14 and Figure

15 can be found on Table B.2 in Appendix B. Looking at the table, also included in the table are








picture.



37

Table B.1. The function of N=2, had a period of 35800 years with an amplitude of

0.49°C. Obviously this period is much larger than the timeframe that we are looking at. This

function has multiple oscillations within the period. The other function N=7 had a period of

1710 years and an amplitude of 0.15°C. This function has a much smaller period which

influences the best fit line to a slightly less degree then the N=2 function. These two functions

set the overall trend of the best fit temperature line. One smaller functions, N=4, played a much

more minor role. With the function N=4, having a period of 233 years and an amplitude of

0.02°C, the amplitudes of this functions cause it only to slightly influence the best fit line.

The larger period function of 35,800 years, is much larger than the time frame

considered, and is more of an artifact generated by the method of analyzing periods. As was

pointed out earlier, Mathematica is simply a software package that is just analyzing the data, not

interpreting it.

38


data from 2000 BCE to 1700 CE. The best fit function to produce the smooth curve in the figure

used only the N=2, N=7and N=4 components. Data adapted from Table B.1 in Appendix B. The

39

coefficients used are in Table B.2 in Appendix B.

As can be seen in Figure 16, the three main functions chosen provide a reasonable




standard deviation of the mean and dividing it by the square root of n. The new data were then

processed using Mathematica and a new set of functions created to best fit this smoothed data

set. This can be seen in Figure 17, with the smoothed data (in blue) and the best fit line in red.



periods of 1880 and 1730 years were calculated.

40



adapted from Table B.1 in Appendix B. The coefficients used are in Table B.3 in Appendix B.

2.3 Conclusion

In conclusion, it is clear that the pattern of climate change that one expects to see from

historical proxy records match reasonably well Equation 3. Also the best fit curve from the

model predicts long term global temperatures reasonably well. The periodic fluctuations seem to

match up to time frequency/periods that have been noted before.

From the 4000 year data sets that included the Scandinavian region, periods of 4450 and

4880 were noticed. When smoothing out the data, smaller periods appeared, from 2100 and

41

3100 year. Other periods appeared, one at 10300 and another at 185,000 years. These two

functions are too large to be determined by the data set, and appear to be “artifact” used as an

adjusting factor to shift the overall function. Mathematica “best fit the function” with the

programming it was given, and generates an output based on that programming. It is possible

that the large period functions (10,300, 185,000, and 35,800 years) are artifacts of the

programming, and not necessarily periods found in nature, unless they are verified by much

longer time periods. When observing the periods from the 4000 year set without the

Scandinavian impact, a 35,800 and 1710 year period stood out. The 35,800 year period wasn’t

sinusoidal and had multiple smaller periods within it. When smoothing out the temperature

reconstruction for the temperature data that did not include the Scandinavian region, the two

functions that fit had periods of 1880 and 1730 years. So, without the Scandinavia data sets, we

see a smaller period around the 1700-1800 year range, which is very similar to a period of 1500

years discussed by Loehle and Singer (Loehle & Singer, 2010).

The functions of Equation 3 between the global temperature reconstruction and the global

temperature reconstruction without the Scandinavian temperature sets varied greatly in that they

were not able to come up with a common periodic function that could be shared between them.

Also the periodic functions of the global temperature reconstruction had smaller periods and

larger amplitudes while the functions in the non Scandinavian global temperature reconstruction

had a much larger period function which had a small amplitude.

42

CHAPTER 3

ANALYSIS OF PROXY DATA SETS OVER 14,000 YEARS OF TEMPERATURE

RECONSTRUCTION

Proxy data were explored over a longer span of time of 14,000 years to test the model for

extended intervals of time. The longer time span allowed other frequency components to be

explored with longer periods of fluctuation identified.

3.2 Introduction

In this chapter, the time interval is extended to 14,000 years of temperature

reconstruction, from 12,000 BCE to 2,000 CE. Temperature reconstruction from 12,000 BCE to

1700 CE, limited to a period before the industrial revolution, was explored to establish a pre-

Industrial Revolution pattern for the temperature. Using these data, a best fit curve that

correlates with the pre-industrial revolution temperatures was derived. The model was then used

to assess wherein the temperature appeared to be headed. The two periods, one not including the

Industrial Revolution (12,000 BCE to 1700 CE) and another including data during the Industrial

Revolution (12,000 BCE to 2000 CE) were tested. These temperature reconstruction from the

data provided best fit curves, in order to compare the temperature trends during the two time

periods.

3.3 Methods/Procedures

Multiple temperature reconstructions were collected from the National Oceanic and

Atmospheric Administration Paleoclimatology Data website (National Oceanic and Atmospheric

Administration, n.d.). Mathematica was used to compile the temperature reconstructions into an

average global reconstruction. In this temperature reconstruction from 12,000 BCE to 2000 CE,

43

3.1 Abstract

144 temperature reconstructions that represent a global distribution were used. This global

distribution can be seen in Figure 18.

Figure 18: A global distribution of sites used to get the temperature reconstructions analyzed in

Chapter 3. Figure created in Mathematica 10 using the latitude and longitude coordinates for

each temperature site. Some sites overlap each other.

44

Figure 19: A graph of the number of sources from

45

Table C.1 that was used for each decade in the temperature reconstruction for the past 14000

years. The number of data sources becomes sparser as proxy data for very ancient times is

collected.


46

Table C.1. In blue is the error bars associated with each decade. The largest error in this


Each set of temperature reconstructions was then averaged within the time frame given,

in this case between 12,000 BCE to 2000 CE, and this average was then subtracted from the set.

The reason the data were averaged over the time frame being studied, was to have the data

centered around 0°C in the each particular time frame. Some of the temperature sets were quite

long, and covered an extensive time period, which included the Ice Age. Since these

temperatures were quite low, when averaged, the temperatures in the time frame of interest

would be well above average, and not centered on the 0 °C mark. By averaging the temperature

sets only around the chosen time period, each set is centered around 0 °C within the time period

that we are looking at.

Once each member of the set had its average subtracted from it, what remains is a set of

data that predicts when the temperature was above average, or in this case above the 0 °C curve,

or when the temperature was cooler than average, or below the 0 °C line.

Once the data is centered, linear interpolation was used to span the gap between some of

the data points to smooth the data.

47

.

Figure 21: The average temperature reconstruction over 14,000 years (Between 12000 BCE to

2000 CE). Data adapted from proxy data sets can be found in

48

Table C.1 in Appendix C.

Once each of the data sets were interpolated, they were then averaged together to find the

general trend of the temperature changes. The average temperature reconstruction for the time

interval 12,000 to 2000 CE can be seen in Figure 21. Note that the temperature has risen from -

1.5° C to 0.5° C in the time interval 11,000 BCE to 4000 BCE. This interval overlaps the Great

Ice Age that occurred in the Northern Hemisphere around 12,000 BCE.

The data of this averaged temperature change was then used to obtain the pattern using

the Equation 3. The data for temperatures after 1700 CE were then removed, to eliminate any

temperatures that could be associated with the industrial revolution. In this case, the temperature

reconstruction between 12,000 BCE and 1700 CE were used for the best fit curve. This

reconstruction can be seen in Figure 22.

49

Figure 22: Temperature reconstruction prior to the Industrial Revolution, between the years

12,000 BCE and 1700 CE. Data used to plot the graph are in

50


The function used for the best fit was Equation 3. Using N=8, a best fit function was

created. The coefficients obtained from the best fit calculations can be found in a Table C.2 in

Appendix C.

Figure 23: A plot of the proxy data and the best fit curve following the trends in the temperature

change between 12,000 BCE to 1700 CE Data were adapted from data sets in Table C.1 in

51

Appendix C. The coefficients of the best fit line (red line) from Equation 3 can be found in


Table C.1. The function of N=2, had a period of 18,300 years with an amplitude of

0.93°C. Obviously this period is much larger than the timeframe spanned. Again, this function

may be an “artifact” of the method employed to analyze the data. The other function for N=5

had a period of 4640 years and an amplitude of 0.189°C. This period is similar to the 4450 and

4880 year periods found in the 4,000 year reconstruction. These two functions set the overall

trend of the best fit temperature line. One smaller functions, N=8, played a less important role.

With the function N=8, having a period of 358 years and an amplitude of 0.00950°C, the

amplitudes of this functions cause it to only slightly influence the best fit line.

Figure 24: A graph of temperature proxy data and a best fit curve using Equation 3 for a time

span of 14,000 years. Data were adapted from data sets inTable C.1 in Appendix C. The

coefficients of the best fit line (red line) from Equation 3 can be found Table C.2 in Appendix C.



52

Figure 25: A plot of the experimental data and a best fit curve from the data shown in Figure 24.

The last 4000 years in the overall span of the 14,000 years of temperature reconstruction are

shown to express trends in recent times. Data were adapted from data sets in Table C.1 in

Appendix C. The coefficients of the best fit line (red line) from Equation 3 can be found in


53

A graph of the major functions that influenced the graph was then overlaid on the data set

to get a more basic best fit line.

Figure 26: The average temperature reconstruction for 13700 years with the best fit function

using data from 12000 BCE to 1700 CE. The best fit function to produce the smooth curve in

the figure used only the N=2, N=5and N=8 components. Data adapted from Table C.1 in

Appendix C. The coefficients used are in Table C.2 in Appendix C.

54










periods of 744, 1120, 1370, 1660, 4550, and 17600 years were calculated.

55



adapted from Table C.1 in Appendix C. The coefficients used are in Table C.3 in Appendix C.

56

3.4 Results

The result of the best fit equation was then compared to the temperature reconstruction

that is shown in Figure 21. In Figure 23, we can see the best fit line and how well it compares to

the temperature change prior to the industrial revolution. If the temperature for the years

between 1700 CE and 2000 CE are added to the set, it can be seen how the best fit function

predicts temperature for the recent years. As can be seen in Figure 24, the best fit model predicts

that the temperature is still declining, while the average temperature reconstruction spiked. If the

temperature is limited to the last 4000 years as can be seen in Figure 25, the rise in the trend of

the temperature reconstruction compared to the best fit function can be seen more clearly.

3.5 Temperature Reconstruction without Using the Scandinavian Data Sets

As was seen in chapter 2, the question arises, with the influence of multiple sources from

the Scandinavian region added in, how that would affect the results and skew the data so the

model will favor that region. In this section the Scandinavia data sets of temperature sources

were removed and the limited data reevaluated. The following analysis includes only the global

data set without the Scandinavian data sets.

Specifically, the data sets from Finland, Sweden, and Norway were not included in the

following analysis. The 93 sets of sources from which the data sets were obtained can be found

in Table D. in Appendix D

57

Figure 28: A map of the global distribution of temperature reconstruction locations used in

analyzing a 14,000 year span without using the data sources in the Scandinavian region. Figure

28 was created in Mathematica 10 using the latitude and longitude coordinates for each

temperature site. Some sites overlap each other. The two sites at the bottom of the figure are

from Antarctica.

58

Figure 29: A graph of the number of sources from Table D.1 that was used for each decade in the

temperature reconstruction for the past 14,000 years. The number of data sources becomes

sparser as proxy data for very ancient times is collected.

A new combination of data that doesn’t include the large set of data from the Scandinavia

region was considered, the data was analyzed again in order to obtain a temperature

reconstruction without the Scandinavian data sets. These results are plotted in Figure 30.

59


Table D.1. In blue is the error bars associated with each decade. The largest error in this


60

Figure 31: Temperature Reconstruction without using data sources from the Scandinavian

countries. Data used to plot the graph are adapted from Table D.1 in Appendix D

The changes are more pronounced, but otherwise the trends are very similar to the

temperature reconstruction which included the Scandinavia sources. Using Equation 3, a best fit

curve using only the data from 12000 BCE to 1700 CE was obtained and plotted in Figure 32.

61

Figure 32: A plot of the raw data for the global proxy data sets between 12,000 BCE and 1700

CE and the profile predicted by using Equation 3. The data sources used in the reconstruction are

listed in Table D.1 in Appendix D.

When the best fit function is extended to the year 2000 CE, it can be seen that the model

is actually higher than the proxy temperature reconstruction since the best fit model just follows

the general trends of the proxy data and not the smaller fluctuations. This can be seen in Figure

33.

62

Figure 33: A reconstruction of temperature over the time period 12000 BCE to 2000 CE. The

black curve is the raw data. The smooth red curve is the temperature trend predicted by Equation

3. The data sets used in the reconstruction are listed in Table D.1 in Appendix D. The

coefficients for the best fit line are in Table D.2 in Appendix D.

The coefficients in Equation 3 that created the curve shown in Figure 32 and Figure 33

can be found in Table D.2 in Appendix D. Looking at the table, also included in the table are the

period (in years) and the amplitude (in °C) that corresponds with each independent function.

Most of these functions make only minor contributions to the curve, with amplitudes as small as

0.002 °C and can be ignored as too small to matter in understanding the big picture of

temperature change.



63


used to get the best fit line, periods that were shorter than ten years were deemed insignificant to

the overall picture.

The functions that correspond to N=3, N=8, and N=6 were the only ones assumed to play

a major role in best fitting the temperature changes from the sources in Table D.1. The function

of N=3, had a period of 67,200 years with an amplitude of 2.09°C. Obviously this period is much

larger than the time frame considered and, generally, is responsible for the overarching shape of

the graph. The other function N=8 had a period of 4490 years with an amplitude of 0.182°C.

This function has a much smaller period which influences the best fit line to a slightly less

degree than the N=3 function. The third function N=6, had a period of 1390 years with an

amplitude of 0.053°C.

The larger period functions of 67,200 years, is much larger than the time frame observed,

and may be more of an artifact generated by the method of analyzing periods Mathematica is

simply a software package that is just analyzing the data as it was programmed; not providing the

meaning behind the numbers it has been given.

64

Figure 34: The average temperature reconstruction for 13700 years with the best fit function

using data from 12000 BCE to 1700 CE. The best fit function to produce the smooth curve in

the figure used only the N=3, N=8 and N=6 components. Data adapted from Table D.1 in

Appendix D. The coefficients used are in Table D.2 in Appendix D.








65



periods of 929, 1120, 9310, and 35100 years were calculated. The functions of 9310 and 35100

years have fluctuations within the periods. The 35,100 year period may be an artifact created by

Mathematica in its process to best fit the data given, and should not be considered a period,

unless it appears in the longer time frames.



adapted from Table D.1 in Appendix D. The coefficients used are in Table D.3 in Appendix D.

66

3.6 Conclusion/Discussion

The proxy temperature data for the past 14,000 years show periods of rising and

decreasing temperatures over the span of time.

The coefficients to Equation 3 that created the function shown in Figure 32 and Figure 33

can be found on Table D.2 in Appendix D. Looking at the table, also included in the table are








picture.

With this in mind, the functions corresponding to N=2 and N=5 were the only two in


67

Table C.1. The function corresponding to N=2, had a period of 18,300 years with an

amplitude of 1.86°C. The large period function of 18,300 can be considered to be an artifact of

the programming, and not necessarily a period found in nature, unless it is observed in data that

covers a significantly longer period than 18,300 years.

The other function N=5 had a period of 4643 years and an amplitude of 0.38°C. These

two functions set the overall trend of the best fit temperature line. Other best fit functions had

periods of 20 years to 40 years which was deemed too small to be a result of the temperature

reconstruction and considered just noise. The other two functions has longer periods, in the

hundreds of years, but the amplitudes to these functions were so minor (0.019°C to 0.002°C) that

they were considered insignificant also.

When considering the case where the Scandinavian data was taken out of the global

temperature reconstruction, the functions that correspond to N=3, N=8, and N=6 were the only

three in which the function played a major role in best fitting the temperature changes from the

68

sources in Table D.. Periods of 67,200, 4490, and 1390 years were observed. It is interesting to

note that a period around 4500 – 4600 is found in both the temperature reconstruction with the

Scandinavian data and the temperature reconstruction without the Scandinavian data.

When you compare the periods calculated with a more smoothed data set, we see periods

in the low to mid thousand years (1120, 1370, and 1660 year with the Scandinavia set and 929

and 1120 with the non Scandinavian set) which have similar periods to a 1500 year period

proposed by Loehle (Loehle & Singer, 2010). Also larger periods were observed, such as a

17600 year period with the Scandinavian data set and a 35,100 year period with the non-

Scandinavian data set. These periods were not so much periodic fluctuations, but rather a basic

function that followed the data as it went from the last ice age and got warmer, and which the

other functions were fluctuating on.

Although the periods of 18,300, 17,600, 67,200 and 35,100 years appear in the 14,000

year span of time, they can be tentatively assumed to be artifacts of the programming, and not

necessarily periods found in nature, unless verified by longer time spans.

69

CHAPTER 4

EVALUATION AN EXTENDED PERIOD OF TIME FOR 1,000,000 YEARS

OF TEMPERATURE CHANGE RECONSTRUCTION

4.1 Abstract

Proxy data were explored over a longer span of time of 1,000,000 years to test the model

for extended intervals of time and to test for any long-term periodic/quasiperiodic changes. The

longer time span allowed other frequency components to be explored with longer periods of

fluctuation identified. In the frame of 1,000,000 years a period of 100,000 years indicated

“peaks” in the data sets explored.

4.2 Introduction

In this chapter proxy data are analyzed that cover a one million year period time span,

from 1,000,000 BCE to 0 CE. The data sets are analyzed to search for the 100,000 year periodic

fluctuations. Multiple sets of data were used to create an average temperature reconstruction.

All proxy data temperate reconstructions that were available in the literature that covered

140,000 years or more were tabulated. These data sets were then used to create a best fit function

using Equation 3, to best represent the temperature change.



study. There were 16 sources that fit the criteria. These data sets were used for this temperature

reconstruction and are listed in Appendix E. As one can see in Figure 36 the different

temperature reconstruction sources represent a global distribution that takes into account a

reconstruction on each continent.

70

The temperature reconstructions for multiple regions were collected from the National

Oceanic and Atmospheric Administration Paleoclimatology Data website (National Oceanic and

Atmospheric Administration, n.d.). The data were then analyzed using Mathematica wherein the

average of the data sets was calculated. The average is then subtracted from the data set to set all

the data sets around zero. Before this, some data sets were centered around 24°C while others

are centered around 3°C. Comparing these data sets does not give a full understanding of the

temperature changes between them. So, by subtracting the average of the temperature for the

data set, the different data sets are referenced to the same level, namely around the 0°C mark,

with temperatures larger than 0°C being above average and temperatures smaller than the

average as below average. Once this data is centered on 0°C, linear interpolation was used to fill

in the interval between some data points. This was done using Equation 2.

71

Figure 36: A map of the data sites used in creating the one million year temperature

reconstruction. Figure created in Mathematica 10 using the latitude and longitude coordinates

for each temperature site. Some sites overlap each other.

72

Figure 37: A graph of the number of sources from Table E.1 that was used for each decade in the

temperature reconstruction for the past one million years. The number of data sources becomes sparsermillion years. The number of data sources becomes sparser as proxy data for very ancient times is collected.

73


Table E.1. In blue is the error bars associated with each decade. The largest error in this


74

The temperature reconstructions were averaged and the average was then subtracted from

each member of the temperature set. This procedures sets each temperature to be centered

around 0°C, where positive temperatures are just warmer periods on average, while negative

temperature are cooler than average. When the temperatures were centered on 0 °C, linear

interpolation was used to fill in the temperature gaps between some of the different temperature

points. The different temperature sets were averaged together to get a global temperature

reconstruction. A graph of the temperature reconstruction, from 1,000,000 BCE to 0 CE, can be

seen in Figure 39.

75

Figure 39: A graph of a time period of one million years of temperature reconstruction based on

16 localized temperature sets. Data were adapted from data sets in Table E.1 in Appendix E.

76

As can be seen in the previous figure, when 16 sources of data are combined, the

following results are obtained. From the graph, it can be seen that the patterns show “dips” and

“peaks” over time. These changes correspond to events in recorded history. For example, we

can see the global cooling periods that correspond to the glacial ice ages, and the warm periods

between them.

An overview of the data plotted in Figure 39 shows quasiperiodic fluctuations with a

period of about 100,000 years and that the temperatures can fluctuate as much as 4°C, especially

in the last 400,000 years. Patterns in this graph raised the question of whether there was periodic

and or quasi periodic fluctuations in the global temperature. As can be seen, the last 400,000

years show a very distinctive pattern.

To model the changes in the temperature and obtain a best fit pattern a sinusoidal

function with frequency modulation as a basis was chosen. The equation used is given by

Equation 3. The number of cosine components was limited to N=12 frequency modulated

sinusoidal functions to best fit the climate reconstruction. In Equation 3, an and cn are amplitudes

for the fluctuations, bn and dn are the phases of each time event. The variables en are the

modulation levels.

4.3 Results

In this chapter it was shown that for the time interval of 1,000,000 years one can identify

about ten “peaks” on average. Equation 3 provides a best fit model that shows periodic changes

over a million years with a period of about 100,000 years.

77

Figure 40: The temperature reconstruction for a 1 million year time period and the best fit line

created using Equation 3. Data were adapted from data sets in Table E.1 in Appendix E. The

coefficients of the best fit line (red line) from Equation 3 can be found Table E.2 in Appendix E.

78

Figure 41: The average temperature reconstruction for 1,000,000 years with the best fit function

using data from 1,000,000 BCE to 0 CE. The best fit function to produce the smooth curve in

the figure used only the N=1 and N=7 components. Data adapted from Table E.1 in Appendix E.

The coefficients used are in Table E.2 in Appendix E.



the dominant periods and recalculate the best fit curve. In this case, every 10,000 years was

averaged and centered in that time span. The standard error of the mean was then calculated by

taking the standard deviation of the mean and dividing it by the square root of n. The new data

79

were then processed using Mathematica and a new set of functions created to best fit this

smoothed data set. This can be seen in Figure 42, with the smoothed data (in blue) and the best

fit line in red.



periods of 87400, 100000, 103000 years were calculated

Figure 42: The smoothed temperature reconstruction for 1,000,000 years with the best fit

function using data from 1000000 BCE to 0 CE. The error bars are the standard error of the

mean. Data adapted from Table E.1 in Appendix E. The coefficients used are in Table E.3 in

Appendix E.

80


Figure 40 shows a plot of the proxy temperature data with the best fit curve through the

data produced by Equation 3. As we can see, the best fit model shows the 100,000 year periods

of the glacial cycle, and fits well with the data in the time periods between 800,000 BCE and

600,000 BCE, where the temperature models did not have a defined peak.

The coefficients to Equation 3 that created the function you see in Figure 40 can be found

on Table E.2 in Appendix E. Looking at the table, also included in the table are the period (in

years) and the amplitude (in °C) that corresponds with each independent function. Most of these

functions are very minor, with amplitudes as small as 0.0007 °C and can be ignored as too minor

to matter in understanding the big picture of temperature change.





picture.

With this in mind, the functions of N=1 and N=7 were the only two in which the function

played a major role in best fitting the temperature changes from the sources in Table E.1

The function of N=1, had a period of 91800 years with an amplitude 0.9°C.

81

The other function N=7 had a period of 106000 years and an amplitude of 1.04°C. These two

functions set the overall trend of the best fit temperature line. The functions N=4 and N=5 were

the next biggest influences to the best fit line, with amplitudes at 0.5°C and 0.8°C respectively.

But since the periods of these functions were 5.3 years and 25 years respectfully, they were

deemed unimportant in the overall trend of the temperature change since their periods were too

small.

With the smoothed data, set at 10,000 year intervals, three functions showed a similar

pattern to the non smoothed data. Periods of 87,400, 100,000, and 103,000 years were calculated

using the smoothed data. The 87,400 year period is very similar to the 91800 period calculated

using the first method. The 100,000 and 103,000 year periods again were very similar to the

previously calculated 106,000 year period. Both of these are similar to the 100,000-105,000 year

period calculated by Milankovich ( ilankovi , 1941).

82

CHAPTER 5

EVALUATION OF AN EXTENDED PERIOD OF TIME OF 3,000,000 YEARS OF

CLIMATE TEMPERATURE RECONSTRUCTION

5.1 Abstract

Proxy data were explored over a longer span of time of 3,000,000 years to test the model

for extended intervals of time and to test for any long-term periodic/quasiperiodic changes. The

longer time span allowed other frequency components to be explored with longer periods of

fluctuation identified.

5.2 Introduction

In this chapter proxy data are analyzed that cover a 3 million year period time span, from

3,000,000 BCE to 0 CE. The data sets are analyzed to search for the 100,000 year periodic

fluctuations. Multiple sets of data were used to create an average temperature reconstruction.

Proxy data temperate reconstructions that were available in the literature that covered 350,000

years or more were tabulated. These data sets were then used to create a best fit function using

Equation 3, to best represent the temperature change.



study. There were 13 sources that fit the criteria. These data sets were used for this temperature

reconstruction and are listed on Table F.1 in Appendix F. As one can see in Figure 43, the

different temperature reconstruction sources represent a global distribution that takes into account

a reconstruction on each continent.

83

The temperature reconstructions for multiple regions were collected from the National

Oceanic and Atmospheric Administration Paleoclimatology Data website (National Oceanic and

Atmospheric Administration, n.d.). The data were then analyzed using Mathematica wherein the

average of the data sets was calculated. The average is then subtracted from the data set to set all

the data sets around zero. Before this, some data sets were centered around 24°C while others

are centered around 3°C. Comparing these data sets does not give a full understanding of the

temperature changes between them. So, by subtracting the average of the temperature for the

data set, the different data sets are referenced to the same level, namely around the 0°C mark,

with temperatures larger than 0°C being above average and temperatures smaller than the

average as below average. Once this data is centered on 0°C, linear interpolation was used to fill

in the interval between some data points. This was done using Equation 2.

84

Figure 43: A map of the location of data sources used in the 3 million year proxy temperature

data analysis. These sources can be found in Table F.1 in Appendix F. Figure created in

Mathematica 10 using the latitude and longitude coordinates for each temperature site. Some sites

overlap each other.

85

Figure 44: A graph of the number of sources from Table F.1 in Appendix F that was used for each

decade in the temperature reconstruction for the past 3 million years. The number of data sources

becomes sparser as proxy data for very ancient times is collected.

86

Figure 45: The graph of the averaged temperature reconstruction (in black) from the sources in

Table F.1. In blue is the error bars associated with each decade. The largest error in this


87

The temperature reconstructions were averaged and the average was then subtracted from

each member of the temperature set. This procedures sets each temperature to be centered

around 0°C, where positive temperatures are just warmer periods on average, while negative

temperature are cooler than average. When the temperatures were centered on 0 °C, linear

interpolation was used to fill in the temperature gaps between some of the different temperature

points. The different temperature sets were averaged together to get a global temperature

reconstruction. A graph of the temperature reconstruction, from 3,000,000 BCE to 0 CE, can be

seen in Figure 46.

88

Figure 46: A plot of the proxy temperature data for a span of 3 Million years. Data were adapted

from data sets in Table F.1 in Appendix F.

89

As can be seen in the previous figure, when 13 sources of data are combined, the

following results are obtained. From the graph, it can be seen that the patterns show “dips” and

“peaks” over time. These changes correspond to events in recorded history, with global cooling

periods that correspond to the glacial ice ages, and the warm periods between them.

An overview of the data plotted in Figure 46 shows quasiperiodic fluctuations with a

period of about 100,000 years and that the temperatures can fluctuate as much as 4°C, especially

in the last 400,000 years.

To model the changes in the temperature and obtain a best fit pattern a sinusoidal

function with frequency modulation as a basis was chosen. The equation used is given by

Equation 3. The number of cosine components was limited to N=8 frequency modulated

sinusoidal functions to best fit the climate reconstruction. In Equation 3, an and cn are amplitudes

for the fluctuations, bn and dn are the phases of each time event. The variables en are the

modulation levels.

5.4 Results

In this chapter it was shown that for the time interval of 3,000,000 years one can identify

about 26 “peaks” on average. Equation 3 provides a best fit model that shows periodic changes

over three million years with a period of about 100,000 years.

90

Figure 47: Data were adapted from data sets in Table F.1 in Appendix F. The coefficients of the

Tbest fit line (red line) from Equation 3 can be found Table F.2 in Appendix F.

91

Figure 48: The average temperature reconstruction for 3,000,000 years with the best fit function

using data from 3,000,000 BCE to 0 CE. The best fit function to produce the smooth curve in

the figure used only the N=3 and N=7 components. Data adapted from Table F.1 in Appendix F.

The coefficients used are in Table F.2 in Appendix F.



the dominant periods and recalculate the best fit curve. In this case, every 10,000 years was

averaged and centered in that time span. The standard error of the mean was then calculated by

taking the standard deviation of the mean and dividing it by the square root of n. The new data

92

were then processed using Mathematica and a new set of functions created to best fit this

smoothed data set. This can be seen in Figure 49, with the smoothed data (in blue) and the best

fit line in red.



periods of 107000, 108000, 258000, and 4720000 years were calculated. It should be noted that

the periods 258000 and 4720000 have fluctuations within the periods.

Figure 49: The smoothed temperature reconstruction for 3,000,000 years with the best fit

function using data from 3000000 BCE to 0 CE. The error bars are the standard error of the

mean. Data adapted from Table F.1 in Appendix F. The coefficients used are in Table F.3 in

Appendix F.

93


The coefficients to Equation 3 that created the function you see in Figure 47 can be found

on Table F.2 in Appendix F. Looking at the table, also included in the table are the period (in

years) and the amplitude (in °C) that corresponds with each independent function. Most of these

functions are very minor, with amplitudes as small as 0.0005 °C and can be ignored as too minor

to matter in understanding the big picture of temperature change.





picture.

With this in mind, the functions of N=3 and N=7 were the only two in which the function

played a major role in best fitting the temperature changes from the sources in Table F.1 The

function of N=3, had a period of 105,000 years with an amplitude 0.5°C.

94

The other function N=7 had a period of 3,190,000 years and an amplitude of 2.75°C. These two

functions set the overall trend of the best fit temperature line. All the other functions that were

calculated had periods and amplitudes that were too small to play a major role in best fitting of

the temperature reconstruction.

When looking at the smoothed out data sets, four periods appeared. The 107,000 and

108,000 year periods and very similar to the 105,000 noticed in the non-smoothed set. These are

periods already predicted by Milankovich ( ilankovi , 1941). The other periods seen were a

258,000 and a 4,720,000 year period. Since these periods fluctuated within their period, they

were not sinusoidal, and thus are larger then what is expected. Smaller fluctuations within the

258,000 year period are similar to the 100,000 year period. The cycle of 4,720,000 years needs

to be further explored over significantly longer time periods to determine if it is an artifact of the

programming, or one of the periods found in nature.

95

CHAPTER 6

SUMMARY, CONCLUSIONS AND PREDICTIONS FOR FUTURE WORK

In this investigation more than 240 proxy temperature data sets were analyzed to

determine what periodic/quasi-periodic patterns are expressed. All of the data sets that are

currently available on select websites and from specific investigations were analyzed in this

work. The types of data sets chosen for study are summarized in Chapter 1. The data sets

obtained by a number of different methods as described in Chapter 1 were averaged and “zero

set” to test for temperature changes in the positive or negative direction around this average

temperature. This procedure allowed all changes in temperature to be referenced to the same

base. Absolute temperatures obtained by each investigator and deposited in the archives could

be reduced to “changes in temperature” by this procedure. Presentation and analysis of the

temperature proxy data are found in Chapters 2, 3, 4 and 5. In each of these chapters select time

periods are considered for analysis. The goal of this paper was to look for periodic and

quasiperiodic patterns in all of the archival proxy temperature data for the Earth that may be

attributed to the “cyclic” changes in the Earth’s orbital and axial motion. It is well known that

the orbit of the Earth is “modulated” by the quasi-periodic motion of the other planets and it is

expected that these fluctuations will manifest themselves in local Earth temperature.

In chapter 2, proxy temperature data sets over a time period of 4000 years covering the

time period 2000 BCE to 2000 CE were explored in to search for short time period patterns of

1000 years or less. A table of these results is provided in Appendix A as Table A.2. The

functions that were found to have periods of less than 1000 years also had very small amplitudes,

which indicated that they were not as important in the overall temperature trend.

96

There are a number of temperature proxy data sets available from the Scandinavian

countries. This large sampling of data produced a large impact on the results. It was decided

that a reasonable approach was to analyze the total data sets by including all of the Scandinavian

data for one procedure and to exclude these data for a second analysis. In Chapter 2, the first

part of the temperature reconstruction, which utilized the Scandinavian data sets, showed two

“periods” of reasonable amplitude, one with a period of 4450 years and another with a period of

4880 years. Even though these two periods were obtained for the best fit curve that is shown in

Figure 6, they do not appear in the analysis of other data sets except in the 14,000 year period.

In the 14,000 year data sets, periods of 4640, 4550, and 4490 years were observed. This is near

the average of 4450 found in the 4000 year data set. It should be noted that in averaging the data

sets some of the signals are smoothed due to the lack of coincidence of time intervals in all of the

data sets. When looking at the smoothed data set for 4000 year with the Scandinavian data, we

also saw a 2120 and 3120 year periods.

When the data sets for 4000 years of time are analyzed without the Scandinavian data

sets, two periodic components arise. One function had a period of 35,800 years and the other had

a period of 1,710 years. The 35,800 year period is much larger than the timeframe that was

analyzed, and needs to be verified in longer time periods to determine if it is an artifact created in

the mathematical process that was used in this paper. In the published literature a cycle near the

1710 year period has been predicted. Loehle has predicted a period of 1450 years obtained from

a limited number of data sets (Loehle & Singer, 2010). One component in the 4,000 year data

set is 1710 years. When adding in the periods found in the smoothed 4000 year non

Scandinavian data sets, periods of 1880 and 1730 years were found. Although these numbers do

not match Loehle’s prediction, it should be noted that Loehle used a limited set of data to make

97

his prediction and the larger value obtained in this work was derived from many more data sets

than Loehle used.

In Chapter 3 the interval of time chosen for investigation was extended to 14,000 years

over the interval of time 12,000 BCE to 2,000 CE to search for longer time patterns that overlap

much of human history. In this study, periods of 18,300 years and 4,650 years were found, as

pointed out above. It should be noted that during part of that period of time there was the last

“Ice Age” during which the temperature rose significantly. The period of 18,300 years is an

artifact of the mathematical process, used to compensate for the quick rise in the temperature

from the last Ice Age, and not a true period of the temperature. When the data set was smoothed,

multiple smaller periods were found, 744, 1120, 1370, and 1660 years. Also a 4550 year period

that is similar to the 4640 year period found without the smoothed data and the 4450 and 4880

year period found with the 4000 year data. A 17,600 year period was also found, similar to the

18,300 year period of the non smoothed data, which is attributed to being an artifact of the

mathematical process and not a true period.

The Scandinavian data was removed from the 14,000 years data and was found to have a

67,200 year period that gave it the overall arch of the graph, and a 4490 and 1390 year period.

The 67,200 period could be an artifact of the mathematical process, used to compensate for the

rise in temperature from the last Ice Age, and doesn’t represent a true period of the temperature.

The 4490 year period is similar to the periods we saw with the 14,000 year with Scandinavian

data set and the 4,000 year Scandinavian data set. The 1390 year period is similar to the other

periods that ranged from 1300 to 1800, and may or may not be related to the period predicted by

Loehle.

98

In Chapter 4 the time span over which proxy temperature data sets were analyzed was

extended to 1,000,000 to determine the periodic/quasi-periodic patterns that help to understand

global temperatures over geological time spans. The data set for 1,000,000 years shows two

periodic components of interest, one of 91,799 years and another of 105,874 years. These are

assumed to compare with the Milankovitch cycles of 95,000 and 105,000 years. When the data

was smoothed out, periods of 87,400, 100,000, and 103,000 years were calculated. These match

the non smoothed periods.

In Chapter 5, the last time frame that was analyzed was a 3 million year time span. Two

periods stood out, one with a 105,000 year period and one with a 3,190,000 year period.

Although only one cycle is represented with this period, it is expected that the cycle will repeat

over longer spans of time. When the data was smoothed, a 107,000 and 108,000 year periods

were found. These are similar to the 105,000 year period found without the smoothed data and

the 105,000 year cycle fond in the 1 million year data. Also a 258,000 and 4,720,000 year

periods were found with the smoothed data. The 258,000 year cycle had fluctuations within it,

and was not a true sinusoidal function. The 4.7 million year period is much larger than the

timeframe that was observed and needs further investigation over longer time periods to

determine if it is a meaningful cycle or an artifact of the mathematical technique used in analysis

of the data.

In a future work, the last 200 years of global temperature are being studied. Recorded

temperature data are available for most of this period of time. Also, satellites have been used to

measure and record solar flux during part of that period of time. Correlation can now be made

with measured Earth temperature and solar flux to determine how the climate of the Earth may

be driven by the solar flux. It is not a goal of this work to establish such correlation.

99

There is some divergence between the temperature trends predicted from the model

developed in this work and measured temperatures. The model should be considered to set a

“baseline” to which the long-term global temperature can be referenced rather than considered to

be able to predict absolute temperature over future time. The model clearly indicates that both

global temperatures have risen over significant periods of time and have fallen over significant

intervals of time. Thus, future temperature events can be assumed to repeat such cycles. In that

only about 200 years of measured temperatures for the globe are available, some caution is

needed in making strong predictions for the near future. It is noted that a period of only 200

years tends to be insignificant over geological times but must be considered in the light of the

span of human life. Immediate concerns that arise from both the predictions from the model and

from measured temperatures is that we must do what we can to try to slow the rising temperature

of the globe. This includes decreasing the release of atmospheric gases that can lead to global

warming as well as increasing the planting of trees and the reduction of deforestation.

The goal of this work has been satisfied in that a new model for temperature assessment

over time from temperature proxy data has been produced. Although a number of “periodic” and

linear models have been developed and presented in the literature, the algorithm used in this

work for a frequency modulated quasi/periodic function has not been previously reported.

In a future work, the algorithm could be applied to longer time spans in that data have

become available that span up to 65,000,000 years. Data have been gathered by Hanson and

Sato and Zachos. (Hansen & Sato, 2012) and (Zachos, et al., 2001).

100

APPENDIX A

PROXY TEMPERATURE DATA (CHAPTER 2)

101

Table A.1: This table is a list of the proxy temperature data sources used in the 4000 year

temperature reconstruction in Chapter 2. There are 109 sources used in this temperature

reconstruction. Location Type Lat Long Elevation Start

Year1

End

Year2

Citation

Sargasso Sea d18O 33.69 -57.61 --- 3125 25 (Keigwin, 1996)

Caribbean Sea d18O 17.89 -66.6 -349 1817 0 (Nyberg, et al., 2002)

Shihua Cave, China S. Width 39.91 116.4 251 2615 -35 (Tan, et al., 2003)

China --3 35 103 --- 1950 -40 (Yang, et al., 2002)

Coast of Africa foram 20.75 -18.58 -2263 23023 88 (deMenocal, et al., 2000)


Timor Sea Mg/Ca,d18O -10.592 125.388 -832 14578 0 (Stott, et al., 2004)

Arafura Sea Mg/Ca,d18O -5.003 133.4448 -2382 14831 139 (Stott, et al., 2004)

Eastern China --4 30 115 --- 1935 -45 (Ge, et al., 2003)

Benguela Upwelling Mg/Ca -25.514 13.027 -1992 21262 0 (Farmer, et al., 2005)

Labrador pollen 54.855 -60.44 --- 11900 100 (Viau & Gajewski, 2009)


Quebec pollen 56.707 -71.658 --- 9000 100 (Viau & Gajewski, 2009)


Central Canada pollen 55.777 -107.25 --- 11900 100 (Viau & Gajewski, 2009)


MacKenzie pollen 62.68 -130.185 --- 11800 100 (Viau & Gajewski, 2009)


Conroy Lake pollen 46.313 -67.846 --- 1915 -18 (Gajewski, 1988)

Great Basin, USA T. Width 38.895 116.02 3 --3 4524 -56 (Salzar, et al., 2013)

Hudson, Lake chironomids 61.9 -145.67 657 9574 -28 (Clegg, et al., 2011)

Moose Lake chironomids 61.37 -143.6 437 6008 -20 (Clegg, et al., 2010)

Quartz Lake chironomids 64.21 -145.81 293 10949 777 (Wooller, et al., 2012)

Rainbow Lake chironomids 60.72 -150.8 63 13506 -54 (Clegg, et al., 2011)

Screaming Lynx Lake chironomids 66.07 -145.4 223 10611 -43 (Clegg, et al., 2011)

Trout Lake chironomids 68.83 -138.75 150 15425 1784 (Irvine, et al., 2012)

Upper Fly Lake pollen 61.07 -138.09 1326 13417 0 (Bunbury & Gajewski, 2009)

Akvaqiak Lake pollen 66.78 -63.95 17 8334 10 (Fréchette & de Vernal, 2009)

HU 90 dinocysts 58.21 -48.37 -3380 11919 1362 (de Vernal, et al., 2013)

HU 91 dinocysts 77.27 -74.33 -663 6756 1549 (Levac, et al., 2001)

Iglutalk Lake pollen 66.14 -66.08 90 10269 -24 (Kerwin, et al., 2004)

Jake Lake pollen 63.67 -65.15 300 8082 -41 (Kerwin, et al., 2004)

Qipisargo Lake pollen 61 -47.75 7 8634 8 (Fréchette & de Vernal, 2009)

Arapisto pollen 60.58 24.8 133 8889 0 (Sarmaja-Korjonen & Seppä,

2007)

Austerkjosen pollen 68.53 17.27 135 8839 19 (Seppä, et al., 2009)

Berkut chironomids 66.35 36.67 25 10118 0 (Ilyashuk, et al., 2005)

Location Type Lat Long Elevation Start

Year 1End

Year 2Citation

Bjørnfjelltjørn pollen 68.43 18.07 510 8860 -45 (Seppä, et al., 2009)

1 Start Years in BP (Before Present, BP = 1950 CE)

2 End Years in BP (Before Present, BP = 1950 CE)

3 Combination of multiple paleoclimate proxy records obtained from ice cores, tree rings, lake sediment, and

historical documents. 4 Reconstruction is based on phonological events (plant budding or flowering dates) from historical and

documentary records, supplemented by winter snow day records from historical documents for some locations. 5 Set based on 3 tree-ring chronology is based on 3 P. longaeva collections: (1) Sheep Mountain, White Mountains,

CA (SHP, 37.52 N. lat., 118.20 W. long.); (2) Mt. Washington, Snake Range, NV (MWA, 38.91 N. lat., 114.31 W.

long.); and (3) Pearl Peak, Ruby Mountains, NV (PRL, 40.23 N. lat., 115.54 W. long.).

102

Brurskardstjørni chironomids 61.42 8.67 1309 10900 -27 (Velle, et al., 2005)

Chuna Lake d18O 67.95 32.48 475 9300 10 (Jones, et al., 2004)

Dalene pollen 58.25 8 40 8940 -57 (Eide, et al., 2006)

Dalmutladdo pollen 69.17 20.72 355 10514 0 (Bjune, et al., 2004)

Flarken pollen 58.55 13.67 108 8982 -50 (Seppä, et al., 2009)

Flotatjønn pollen 59.67 7.55 890 9021 -44 (Seppä, et al., 2009)

Gammelheimvatnet pollen 68.47 17.75 290 8968 -42 (Seppä, et al., 2009)

Gilltjärnen chironomids 60.08 15.83 172 10707 -50 (Antonsson, et al., 2006)

Gilltjärnen pollen 60.08 15.83 172 10707 -50 (Antonsson, et al., 2006)

Lilla Gloppsjön pollen 59.83 16.53 198 8972 0 (Seppä, et al., 2009)

Grostjørna pollen 58.53 7.73 180 8914 -47 (Eide, et al., 2006)

Over Gunnarsfjorden pollen 71.04 28.17 78 9124 -40 (Allen, et al., 2007)


Haugtjern pollen 60.83 10.88 338 8959 -23 (Eide, et al., 2006)

Holebudalen pollen 59.83 6.98 1144 8929 -10 (Velle, et al., 2005)

Holebudalen chironomids 59.83 6.98 1144 8929 -10 (Velle, et al., 2005)

Isbenttjønn pollen 59.77 7.43 787 8890 -44 (Seppä, et al., 2009)

Kinnshaugen pollen 62.02 10.37 591 8829 -23 (Seppä, et al., 2009)

Klotjärnen pollen 61.82 14.58 235 8942 -48 (Seppä, et al., 2009)

KP-2 pollen 68.8 35.32 131 8997 0 (Seppä, et al., 2009)

Laihalampi pollen 61.48 26.07 137 8995 49 (Heikkilä & Seppä, 2003)

Lake 850 diatoms 68.37 19.12 850 9475 -50 (Larocque & Bigler, 2004)

Lake 850 chironomids 68.37 19.12 850 9475 -50 (Larocque & Bigler, 2004)

Litlvatnet pollen 68.52 14.87 106 8780 -45 (Seppä, et al., 2009)

Myrvatnet pollen 68.65 16.38 200 8894 -42 (Seppä, et al., 2009)

Nautajärvi pollen 61.8 24.7 104 8998 51 (Ojala, et al., 2008)

Voulep Njakajaure diatoms 68.33 18.78 409 8855 111 (Bigler, et al., 2006)

Njulla diatoms 68.37 18.7 999 9477 -48 (Bigler, et al., 2003)

Njulla chironomids 68.37 18.7 999 9477 -48 (Bigler, et al., 2003)

Vestre Økjamyrttjørn pollen 59.82 6 570 11532 -51 (Bjune, et al., 2005)

Vestre Økjamyrttjørn chironomids 59.82 6 570 11532 -51 (Bjune, et al., 2005)

Raigastvere pollen 58.58 26.65 53 8923 0 (Seppä & Poska, 2004)

Råtasjøen chironomids 62.27 9.83 1169 10869 -49 (Velle, et al., 2005)

Ruila pollen 59.17 24.43 43 10013 0 (Seppä & Poska, 2004)

Sjuodjijaure pollen 67.37 18.07 826 9360 0 (Rosén, et al., 2001)

Sjuodjijaure chironomids 67.37 18.07 826 9360 0 (Rosén, et al., 2001)

Søylegrotta d18O 66.62 13.68 280 9955 137 (Lauritzen & Lundberg, 1999)

Spåime chironomids 63.12 12.32 887 10470 42 (Hammarlund, et al., 2004)

Svanåvatnet pollen 66.44 14.05 243 8750 -18 (Bjune & Birks, 2008)


Tibetanus pollen 68.33 18.7 560 10238 39 (Hammarlund, et al., 2002)

Toskaljarvi pollen 69.2 21.47 704 8981 -46 (Seppä & Birks, 2002)

Toskaljarvi chironomids 69.2 21.47 704 8981 -46 (Seppä & Birks, 2002)

Tsuolbmajavri diatoms 68.41 22.05 526 10719 0 (Korhola, et al., 2000)

Tsuolbmajavri chironomids 68.41 22.05 526 10719 0 (Korhola, et al., 2000)

Vuoskkujavri pollen 68.33 19.1 348 10214 -42 (Bigler, et al., 2002)

Vuoskkujavri chironomids 68.33 19.1 348 10214 -42 (Bigler, et al., 2002)

JR01 pollen 69.9 -95.07 120 6956 -48 (Zabenskie & Gajewski, 2007)

Lake K2 chironomids 58.73 -65.93 167 6717 -48 (Fallu, et al., 2005)

JR51-GC35 alkenones 67.59 -17.56 -420 10171 62 (Bendle & Rosell-Melé, 2007)

MD95-2011 foram 66.97 7.64 -1048 6026 -45 (Berner, et al., 2010)

MD95-2015 alkenones 58.77 -25.97 --- 10028 725 (Giraudeau, et al., 2000)

MD99-2256 foram 64.3 -24.21 246 11333 -43 (Ólafsdóttir, et al., 2010)


Location Type Lat Long Elevation Start

Year 1End

Year 2Citation

MD99-2269 diatoms 66.85 -20.85 --- 11477 -16 (Justwan, et al., 2008)

103

Troll 28-03 foram 60.87 3.73 -345 9800 240 (Klitgaard-Kristensen, et al.,

2001)

Dolgoe Lake pollen 71.87 127.07 --- 9952 11 (Wolfe, et al., 2000)

PL-96-112 BC dinocysts 71.74 42.61 -286 8561 203 (Voronina, et al., 2001)

Vostok Station deuterium -78 106 3488 422766 0 (Petit, et al., 1999)

Lake Turkana --- 3.76 35.95 360 2528 476 (Berke, et al., 2012)

Laguna Escondida --- -45.52 -71.82 --- 1550 -58 (Elbert, et al., 2013)

Cariaco Basin Mg/Ca 10.72 -64.7 -450 1970 -58 (Wurtzel, et al., 2013)


Core 70 GGC Mg/Ca -3.57 119.38 -482 14300 6 (Linsley, et al., 2010)

Core 70 GGC d18O -3.57 119.38 -482 14200 0 (Linsley, et al., 2010)


Indo-Pacific Warm Pool Mg/Ca -5.2 117.49 -1185 945 109 (Newton, et al., 2006)

PO287-26-1B/3G alkenone 38.56 -9.35 -96 2061 -51 (Eiríksson, et al., 2006)

MD99-2275 diatoms 66.55 -17.70 -410 1949 55 (Eiríksson, et al., 2006)

West of Iceland diatoms 66.63 -23.85 -365 11477 -16 (Justwan, et al., 2008)

Hinterburgsee chironomids 46.72 8.07 1515 12901 -40 (Heiri, et al., 2003)

MD99-2275 alkenone 66.55 -17.7 -470 1949 -51 (Sicre, et al., 2011)

(3.) Regional temperature (July and January) reconstructions based on 117 fossil pollen records.

(4.) Averaged location based on site latitude and Longitude used in this set.

Table A.2: This is a table of the coefficients obtained using Equation 3 to fit the proxy

temperature data. Mathematica was used to generate the best fit curve for the data set covering

4000 years of time.

N an bn cn dn en PERIOD AMPLITUDE

1 4.20 1.63 -0.00869 0.337 0.00880 714 0.0364

2 0.0351 -1.17 0.341 -3.48 0.0267 235 0.0108

3 0.997 3.62 0.00691 -2.34 -0.0486 129 0.00316

4 -8.12 0.404 0.799 -2.81 -0.00141 4450 2.61

5 -2.69 2.15 1.69 0.264 -0.00129 4880 2.56

6 7.10 -0.375 0.00111 1.67 -0.0516 122 0.00288

7 0.00142 1.43 -1.12 1.61 0.128 49.1 0.001265

8 -0.00578 -0.731 -0.307 3.86 -0.198 63.4 0.001165

Table A.3: This is a table of the coefficients obtained using Equation 3 to fit the smoothed proxy


4000 years of time. N an bn cn dn en PERIOD AMPLITUDE

1 -0.0912 1.92 2.45 1.18 -0.00296 2120 0.0912

2 0.0548 0.204 -5.50 0.803 -0.00103 3120 0.0548

3 0.801 1.01 2.63 1.37 -0.0000340 185000 0.801

4 -0.0684 -7.20 12.1 -0.146 0.000611 10300 0.0684

104

APPENDIX B

PROXY TEMPERATURE DATA EXCLUDING SCANDINAVIA DATA (CHAPTER 2)

105

Table B.1: This table is a list of the proxy temperature data sources used in the 4000 year

temperature reconstruction in Chapter 2 not including the Scandinavia data. There are 63 sources

used in this temperature reconstruction. Location Type Lat Long Elevation Start Year 1 End Year 2 Citation

Sargasso Sea d18O 33.69 -57.61 --- 3125 25 (Keigwin, 1996)

Caribbean Sea d18O 17.89 -66.6 -349 1817 0 (Nyberg, et al., 2002)

Shihua Cave, China S. Width 39.91 116.4 251 2615 -35 (Tan, et al., 2003)

China 1 35 103 --- 1950 -40 (Yang, et al., 2002)



Timor Sea Mg/Ca, d18O -10.59 125.388 -832 14578 0 (Stott, et al., 2004)

Arafura Sea Mg/Ca, d18O -5.003 133.4448 -2382 14831 139 (Stott, et al., 2004)

Eastern China 2 30 115 --- 1935 -45 (Ge, et al., 2003)










Conroy Lake pollen 46.313 -67.846 1915 -18 (Gajewski, 1988)

Great Basin, USA T. Width 38.89 3 116.02 3 3 4524 -56 (Salzar, et al., 2013)

Hudson, Lake chironomids 61.9 -145.67 657 9574 -28 (Clegg, et al., 2011)

Moose Lake chironomids 61.37 -143.6 437 6008 -20 (Clegg, et al., 2010)








HU 91 dinocysts 77.27 -74.33 -663 6756 1549 (Levac, et al., 2001)

Iglutalk Lake pollen 66.14 -66.08 90 10269 -24 (Kerwin, et al., 2004)

Jake Lake pollen 63.67 -65.15 300 8082 -41 (Kerwin, et al., 2004)












MD99-2256 Foram 64.3 -24.21 246 11333 -43 (Ólafsdóttir, et al., 2010)

MD99-2264 Foram 66.68 -24.2 235 11503 -24 (Ólafsdóttir, et al., 2010)



2001)



Location Type Lat Long Elevation Start Year 1 End Year 2 Citation



106

Laguna Escondida --- -45.52 -71.82 --- 1550 -58 (Elbert, et al., 2013)






Indo-Pacific Warm Pool Mg/Ca -5.2 117.49 -1185 945 109 (Newton, et al., 2006)

PO287-26-1B/3G alkenones 38.56 -9.35 -96 2061 -51 (Eiríksson, et al., 2006)

MD99-2275 diatoms 66.55 -17.70 -410 1949 55 (Eiríksson, et al., 2006)



MD99-2275 alkenone 66.55 -17.7 -470 1949 -51 (Sicre, et al., 2011)

Table B.2: This is a table of the coefficients obtained using Equation 3 to fit the proxy


4000 years of time.


1 0.00578 0.563 2.51 -0.724 -0.0863 72.8 0.00577

2 0.247 -51.6 52.9 6.19 -0.000175 35800 0.247

3 -0.00721 -7.78 -2.87 1.43 0.173 36.3 0.0721

4 0.635 0.209 -0.112 2.58 0.0270 233 0.0147

5 0.00793 -0.0600 -2.81 1.33 -0.0379 81.7 0.00779

6 -0.00419 -1.48 -3.08 13.0 -0.171 36.8 0.00419

7 0.0733 -1.13 2.24 -4.79 0.00368 1710 0.0732

8 2.54 -4.44 0.00124 0.0493 0.156 40.3 0.00303

Table B.3: This is a table of the coefficients obtained using Equation 3 to fit the smoothed proxy


4000 years of time.


1 12.1 -1.39 -0.0405 -1.79 -0.00334 1880 0.482

2 -2.64 -0.0214 -0.869 0.948 0.00182 1730 0.446

107

APPENDIX C


108

Table C.1: This table is a list of the proxy temperature data sources used in the 14,000 year


reconstruction. Location Type Lat Long Elev. Start

Year 1End

Year 2Citation



Timor Sea Mg/Ca,

d18O

-10.59 125.388 -832 14578 0 (Stott, et al., 2004)

Arafura Sea Mg/Ca,

d18O

-5.003 133.444 -2382 14831 139 (Stott, et al., 2004)

Davao Gulf Mg/Ca,

d18O

6.3 125.83 -2114 14920 13 (Stott, et al., 2004)

Eastern China 4 --- 30 115 --- 1935 -45 (Ge, et al., 2003)


Alaska 6 pollen 65.912 -144.23 --- 25000 100 (Viau, et al., 2008)



Alaska 9 pollen --- 25000 100 (Viau, et al., 2008)















HU84 dinocysts 58.37 -57.51 -2853 8297 1968 (de Vernal, et al., 2001)




LS009 dinocysts 74.19 -81.2 -781 10821 2035 (Ledu, et al., 2010)


Arapisto pollen 60.58 24.8 133 8889 0 (Sarmaja-Korjonen & Seppä,

2007)

Austerkjosen pollen 68.53 17.27 135 8839 19 (Seppä, et al., 2009)

Location Type Lat Long Elev. Start

Year 1End

Year 2Citation

6Mean Annual Temperature based on 45 sites

7 Mean July Temperature based on 45 sites

8 Mean January Temp based on 45 sites

9Temperature for region A: 65-70N; 150-180W

10 Temperature for region B: 65-60N; 125-150W

11 Temperature for region C: 60-65N; 150-180W

12Temperature for region D: 60-65N; 125-150W

109


Bjørnfjelltjørn pollen 68.43 18.07 510 8860 -45 (Seppä, et al., 2009)

Brurskardstjørni chironomids 61.42 8.67 1309 10900 -27 (Velle, et al., 2005)


Dalene pollen 58.25 8 40 8940 -57 (Eide, et al., 2006)

Dalmutladdo pollen 69.17 20.72 355 10514 0 (Bjune, et al., 2004)

Flarken pollen 58.55 13.67 108 8982 -50 (Seppä, et al., 2009)

Flotatjønn pollen 59.67 7.55 890 9021 -44 (Seppä, et al., 2009)

Gammelheimvatnet pollen 68.47 17.75 290 8968 -42 (Seppä, et al., 2009)

Gilltjärnen chironomids 60.08 15.83 172 10707 -50 (Antonsson, et al., 2006)

Gilltjärnen pollen 60.08 15.83 172 10707 -50 (Antonsson, et al., 2006)

Lilla Gloppsjön pollen 59.83 16.53 198 8972 0 (Seppä, et al., 2009)

Grostjørna pollen 58.53 7.73 180 8914 -47 (Eide, et al., 2006)


Haugtjern pollen 60.83 10.88 338 8959 -23 (Eide, et al., 2006)

Holebudalen pollen 59.83 6.98 1144 8929 -10 (Velle, et al., 2005)

Holebudalen chironomids 59.83 6.98 1144 8929 -10 (Velle, et al., 2005)

Isbenttjønn pollen 59.77 7.43 787 8890 -44 (Seppä, et al., 2009)

Kinnshaugen pollen 62.02 10.37 591 8829 -23 (Seppä, et al., 2009)

Klotjärnen pollen 61.82 14.58 235 8942 -48 (Seppä, et al., 2009)


Laihalampi pollen 61.48 26.07 137 8995 49 (Heikkilä & Seppä, 2003)

Lake 850 diatoms 68.37 19.12 850 9475 -50 (Larocque & Bigler, 2004)

Lake 850 chironomids 68.37 19.12 850 9475 -50 (Larocque & Bigler, 2004)

Litlvatnet pollen 68.52 14.87 106 8780 -45 (Seppä, et al., 2009)

Myrvatnet pollen 68.65 16.38 200 8894 -42 (Seppä, et al., 2009)

Nautajärvi pollen 61.8 24.7 104 8998 51 (Ojala, et al., 2008)

Voulep Njakajaure diatoms 68.33 18.78 409 8855 111 (Bigler, et al., 2006)

Njulla diatoms 68.37 18.7 999 9477 -48 (Bigler, et al., 2003)

Njulla chironomids 68.37 18.7 999 9477 -48 (Bigler, et al., 2003)

Vestre Økjamyrttjørn pollen 59.82 6 570 11532 -51 (Bjune, et al., 2005)

Vestre Økjamyrttjørn chironomids 59.82 6 570 11532 -51 (Bjune, et al., 2005)


Råtasjøen chironomids 62.27 9.83 1169 10869 -49 (Velle, et al., 2005)


Sjuodjijaure pollen 67.37 18.07 826 9360 0 (Rosén, et al., 2001)

Sjuodjijaure chironomids 67.37 18.07 826 9360 0 (Rosén, et al., 2001)

Søylegrotta d18O 66.62 13.68 280 9955 137 (Lauritzen & Lundberg,

1999)

Spåime chironomids 63.12 12.32 887 10470 42 (Hammarlund, et al., 2004)



Svartkälstjärn pollen 63.35 9.55 183 8938 -37 (Seppä, et al., 2009)

Tiåvatnet pollen 63.05 9.42 464 8988 -50 (Seppä, et al., 2009)

Tibetanus pollen 68.33 18.7 560 10238 39 (Hammarlund, et al., 2002)

Toskaljarvi pollen 69.2 21.47 704 8981 -46 (Seppä & Birks, 2002)

Toskaljarvi chironomids 69.2 21.47 704 8981 -46 (Seppä & Birks, 2002)

Trehörningen pollen 58.55 7 112 8894 0 (Antonsson & Seppä, 2007)

Trettetjørn pollen 60.72 22.05 810 8527 -52 (Bjune, et al., 2005)

Tsuolbmajavri diatoms 68.41 22.05 526 10719 0 (Korhola, et al., 2000)

Tsuolbmajavri chironomids 68.41 22.05 526 10719 0 (Korhola, et al., 2000)

Tsuolbmajavri pollen 68.41 22.05 526 10719 0 (Korhola, et al., 2000)

Vuoskkujavri pollen 68.33 19.1 348 10214 -42 (Bigler, et al., 2002)

Vuoskkujavri chironomids 68.33 19.1 348 10214 -42 (Bigler, et al., 2002)

Yarnishnoe pollen 69.07 36.07 54 12484 1198 (Seppä, et al., 2008)

2005-804-006 dinocysts 68.99 -106.57 -118 7731 1024 (Ledu, et al., 2010)


Year 1End

Year 2Citation

110


Ennadai Lake pollen 61.17 -100.92 168 7100 354 (Bender, et al., 1967)


JR01 pollen 69.9 -95.07 120 6956 -48 (Zabenskie & Gajewski,

2007)


B997-321 d18O 66.53 -21.5 --- 6128 177 (Smith, et al., 2005)

GIK23258-2/3 d18O 75 14 -1768 13987 0 (Sarnthein, et al., 2003)

T88-2 foram 71.99 14.36 --- 13355 -50 (Hald, et al., 2007)

JM96-1207 dinocysts 68.1 -29.35 -404 10589 457 (Solignac, et al., 2006)


JR51-GC35 alkenones 67.59 -17.56 -420 10171 62 (Bendle & Rosell-Melé,

2007)

LO09-14 diatoms 58.94 -30.41 --- 10976 368 (Berner, et al., 2008)




Malangenfjord d18O,foram 69.5 18.39 -213 7891 1580 (Husum & Hald, 2004)


MD95-2015 alkenone 58.77 -25.97 --- 10028 725 (Giraudeau, et al., 2000)




ODP 684, Bjørn Drift d18O 61 -25 -1648 10423 554 (Came, et al., 2007)


2001)

Troll 28-03 d18O 60.87 3.73 -345 9800 240 (Klitgaard-Kristensen, et al.,

2001)





Dome Fuji Station d18O -77.32 38.7 3810 337650 -47 (Uemura, et al., 2012)


Lake Tanganyika --- -6.75 29.97 --- 59454 1313 (Tierney, et al., 2008)

Sacred Lake --- 0.08 37.53 2350 47730 63 (Loomis, et al., 2012)

ME0005A-27JC alkenone -1.853 -82.79 -2203 29680 1210 (Dubois, et al., 2009)

ME0005A-43JC alkenone 7.86 -83.61 -1368 24920 2310 (Dubois, et al., 2009)

Coast of Brazil Mg/Ca -4.61 -36.64 -830 20740 170 (Weldeab, et al., 2006)

Alboran Sea alkenone 36.14 -2.62 -1841 51930 1008 (Cacho, et al., 1999)

ODP 658C alkenone 20.75 -18.58 -2263 83906 46 (Zhao, et al., 1995)

ODP 658C alkenone 20.75 -18.58 -2263 83906 46 (Zhao, et al., 1995)

MD85674 alkenone 3.18 50.43 4875 149200 0 (Bard, et al., 1997)

Congo River Basin foram -5.59 11.22 -962 25790 30 (Weijers, et al., 2007)

GeoB 1023-4 alkenone -17.16 11.01 1978 21190 310 (Kim, et al., 2002)

ODP-1078C alkenone -11.92 13.4 -426 24740 20 (Kim, et al., 2003)




MD97-2120 alkenone 13 -45.53 174.93 -1210 156360 1951 (Pahnke & Sachs, 2006)


Location Type Lat Long Elev. Start End Citation

13 Sea surface temperature from Alkenone using (Prahl et al. 88) Method

14Sea surface temperature from Alkenone using (Sikes and Volkman 93) Method

111

Year 1 Year 2

Sea of Okhotsk alkenone 49.37 153.02 -1821 30450 1590 (Harada, et al., 2004)

GeoB 10029-4 d18O -1.49 100.13 -964 40120 0 (Mohtadi, et al., 2010)


CH84-04 d18O 36.77 142.22 -2630 19126 2201 (Oba & Murayama, 2004)

South China Sea Mg/Ca 19.58 117.63 -3175 143000 1940 (Oppo & Sun, 2005)

A7 Mg/Ca 28.01 127.02 -1264 18193 1367 (Sun, et al., 2005)



HLY0501-05 dinocysts 72.69 -157.52 -415 8209 225 (McKay, et al., 2008)

GIK23258-2/3 foram 75 14 -1768 13987 0 (Sarnthein, et al., 2003)

Table C.2: This is a table of the coefficients obtained using Equation 3 to fit the proxy


14,000 years of time.


1 0.00520 -0.479 -0.467 4.92 0.103 122 0.00108

2 3.48 -0.557 -0.530 -4.43 -0.000342 18300 0.930

3 0.865 3.57 -0.00209 -3.90 0.160 39.2 0.000743

4 -0.849 0.272 -0.00237 1.41 0.153 41.1 0.000541

5 1.31 -2.48 -0.236 1.63 0.00135 4640 0.189

6 -0.000798 1.24 1.29 -4.81 -0.227 27.7 0.000726

7 0.541 -3.65 -0.00216 -1.51 -0.176 35.8 0.000568

8 -0.0825 0.205 0.487 0.843 -0.0175 358 0.00950

Table C.3: This is a table of the coefficients obtained using Equation 3 to fit the smoothed proxy


14000 years of time.


1 -0.352 1.29 -0.120 1.49 -0.00379 1660 0.0405

2 -0.0781 0.141 -1.01 -0.871 0.00459 1370 0.0232

3 -0.264 -1.16 -0.796 1.42 -0.00138 4550 0.173

4 -0.960 0.471 -3.34 0.617 -0.000162 17600 0.530

5 0.303 -0.209 0.427 -0.919 -0.00561 1120 0.0296

6 0.0896 -2.79 0.645 -0.0760 -0.00845 744 0.0205

112

APPENDIX D

PROXY TEMPERATURE DATA EXCLUDING SCANDINAVIA DATA (CHAPTER 3)

113

Table D.1: This table is a list of the proxy temperature data sources used in the 14,000 year

temperature reconstruction without the Scandinavian data sets in Chapter 3. There are 93 sources

used in this temperature reconstruction. Location Type Lat Long Elev. Start

Year 1End

Year 2Citation

Coast of Africa foram 20.75 -18.5833 N/A 23023 88 (deMenocal, et al., 2000)

Coast of Africa foram 20.75 -18.5833 N/A 23023 88 (deMenocal, et al., 2000)

Timor Sea

Mg/Ca,

d18O -10.592 125.3882 -832 14578 0 (Stott, et al., 2004)

Arafura Sea

Mg/Ca,

d18O -5.003 133.4448 -2382 14831 139.34 (Stott, et al., 2004)

Davao Gulf

Mg/Ca,

d18O 6.3 125.83 -2114 14920 13.84 (Stott, et al., 2004)

Eastern China 4 --- 30 115 --- 1935 -45 (Ge, et al., 2003)

Benguela Upwelling Region Mg/Ca -25.5138 13.02783 -1992 21262 0 (Farmer, et al., 2005)























LS009 dinocysts 74.19 -81.2 -781 10821 2035 (Ledu, et al., 2010)







Yarnishnoe pollen 69.07 36.07 54 12484 1198 (Seppä, et al., 2008)








Year 1End

Year 2Citation

B997-321 d18O 66.53 -21.5 --- 6128 177 (Smith, et al., 2005)

114

GIK23258-2/3 d18O 75 14 -1768 13987 0 (Sarnthein, et al., 2003)

T882-2 foram 71.99 14.36 --- 13355 -50 (Hald, et al., 2007)













ODP 684 d18O 61 -25 -1648 10423 554 (Came, et al., 2007)

Troll 28-03 foram 60.87 3.73 -345 9800 240 (Klitgaard-Kristensen, et al., 2001)

Troll 28-03 d18O 60.87 3.73 -345 9800 240 (Klitgaard-Kristensen, et al., 2001)





Dome Fuji Station d18O -77.32 38.7 3810 337650 -47 (Uemura, et al., 2012)


Lake Tanganyika --- -6.7488 29.9703 --- 59454 1313 (Tierney, et al., 2008)

Sacred Lake --- 0.0833 37.5333 2350 47730 63 (Loomis, et al., 2012)

ME0005A-27JC alkenone -1.85333 -82.7867 -2203 29680 1210 (Dubois, et al., 2009)

ME0005A-43JC alkenone 7.855833 -83.6083 -1368 24920 2310 (Dubois, et al., 2009)

Coast of Brazil Mg/Ca -4.61333 -36.6367 -830 20740 170 (Weldeab, et al., 2006)

Alboran Sea Alkenone 36.14333 -2.62167 -1841 51930 1008 (Cacho, et al., 1999)

ODP658C alkenone 20.75 -18.5833 -2263 83906 46 (Zhao, et al., 1995)

MD85674 Alkenone 3.18333 50.43333 4875 149200 0 (Bard, et al., 1997)

Congo River Basin Foram -5.58833 11.22167 -962 25790 30 (Weijers, et al., 2007)

GeoB1023-4 alkenone -17.1583 11.00833 1978 21190 310 (Kim, et al., 2002)

ODP-1078C alkenone -11.92 13.4 -426 24740 20 (Kim, et al., 2003)


Core 70GGC d18O -3.56667 119.3833 -482 14200 0 (Linsley, et al., 2010)

Core 13GGC d18O -7.4 115.2 -594 10550 100 (Linsley, et al., 2010)



Sea of Okhotsk alkenone 49.36667 153.0167 -1821 30450 1590 (Harada, et al., 2004)


GeoB10038-4 d18O -5.9375 103.246 -1819 39590 0 (Mohtadi, et al., 2010)

CH84-04 d18O 36.76667 142.2167 -2630 19126 2201 (Oba & Murayama, 2004)


A7 Mg/Ca 28.01361 127.0164 -1264 18193 1367 (Sun, et al., 2005)


Hinterburgsee chironomids 46.7186 8.0686 1515 129.1 -40 (Heiri, et al., 2003)

HLY0501-05 dinocysts 72.69 -157.52 -415 8209 225 (McKay, et al., 2008)

GIK23258-2/3 foram 75 14 -1768 13987 0 (Sarnthein, et al., 2003)

115

Table D.2: This is a table of the coefficients obtained using Equation 3 to fit the proxy


14,000 years of time.


1 0.0487 -1.74 -0.140 20.0 -0.786 8.00 0.00670

2 -0.00805 -17.6 0.291 -2.60 0.0157 399 0.00221

3 -2.09 1.26 -2.52 9.75 -0.0000935 67200 2.09

4 -0.0270 -0.275 -1.03 4.76 -0.149 42.3 0.00996

5 -0.0746 4.15 -0.189 -29.2 0.222 28.4 0.0119

6 -0.0999 5.39 0.751 12.8 0.00453 1390 0.0530

7 -1.53 -0.0553 0.0677 137 0.179 35.2 0.00576

8 0.549 -1.29 0.353 -1.66 -0.00140 4490 0.182

Table D.3: This is a table of the coefficients obtained using Equation 3 to fit the smoothed proxy


14000 years of time.


1 0.514 -0.657 -0.0948 -0.636 -0.00676 929 0.0297

2 -0.0295 2.19 2.78 3.28 -0.00563 1120 0.0295

3 -3.64 -3.00 -0.943 0.753 -0.000179 35100 0.970

4 -3.73 0.134 0.457 -0.820 0.000675 9310 0.316

116

APPENDIX E


117

Table E.1: This table is a list of the proxy temperature data sources used in the 1,000,000 year


reconstruction. Location Type Lat Long Eleva

tion

Start

Year 1End

Year 2Citation


ODP 871 Mg/Ca 5.55 172.34 -1254 537150 0 (Dyez & Ravelo, 2013)

Dome Fuji Station d18O -77.32 38.7 3810 358950 50 (Uemura, et al., 2012)

NA87-22/NA87-

25/ODP980

foram 55.39 -14.72 -2220 429500 500 (Waelbroeck, et al., 2002)

V19-30/ODP677 foram -1.08 -83.54 -3276 429500 500 (Waelbroeck, et al., 2002)

MD94-101 foram -42.5 79.42 -2920 429380 5000 (Waelbroeck, et al., 2002)

Atlantic Ocean alkenone -31.47 15.32 -1372 1517112 497630 (McClymont, et al., 2005)

MD06-3018/ Coral Sea foram -23 166.15 -2470 1543500 0 (Russon, et al., 2010)

Ontong Java Plateau Mg/Ca 0.32 159.36 -2520 470000 4300 (Lea, et al., 2000)

MD97-2120 alkenone15 -45.53 174.93 -1210 156360 1951 (Pahnke & Sachs, 2006)

MD97-2120 alkenone16 -45.53 174.93 -1210 156360 1951 (Pahnke & Sachs, 2006)


ODP 722 / Arabian Sea alkenone 16.62 59.80 -2022 3330480 7110 (Herbert, et al., 2010)

East Atlantic Ocean Mg/Ca -5.77 -10.75 -3122 271000 1000 (Nuernberg, et al., 2000)

ODP 882 / Pacific O. alkenone 50.35 167.58 -3244 3638880 4270 (Martínez-Garcia, et al., 2010)

ODP 1090 / Atlantic O. alkenone -42.91 8.9 -3702 3642410 0 (Martínez-Garcia, et al., 2010)

Table E.2: This is a table of the coefficients obtained using Equation 3 to fit the proxy


1,000,000 years of time.


1 0.639 -1.34 0.822 -3.87 0.0000684 91800 0.455

2 0.0186 1.27 0.141 60.3 0.183 34.3 0.00249

3 -0.00229 -14.1 0.158 46.8 -8.15 0.771 0.000361

4 0.252 4.47 -1.66 -276 -1.19 5.26 0.251

5 0.475 -10.7 1.10 34.6 -0.251 25.0 0.407

6 0.0300 -14.6 -0.302 125 2.41 2.61 0.00813

7 -0.799 1.92 0.762 0.596 -0.0000593 106000 0.519

8 -0.0248 15.4 0.314 -41.8 8.02 0.784 0.00254

9 0.266 -42.0 0.136 4.34 -6.01 1.05 0.0326

10 0.00761 -51.4 0.151 239 -8.44 0.745 0.00103

11 -0.128 0.787 -0.0212 -48.7 -0.0214 294 0.00192

12 0.0345 -34.3 0.846 196 1.16 3.76 0.0100

15 SST Reconstruction based on Prahl et al. 1988

16 SST Reconstruction based on Sikes and Volkman 1993

118

Table E.3: This is a table of the coefficients obtained using Equation 3 to fit the smoothed

proxy temperature data. Mathematica was used to generate the best fit curve for the data set

covering 1,000,000 years of time.


1 -8.82 -2.84 0.177 1.26 0.0000719 87400 0.461

2 1.31 -1.49 -1.18 -3.99 0.0000626 100000 1.21

3 27.8 1.88 -0.0544 1.20 0.0000612 103000 1.44

119

APPENDIX F


120

Table F.1: This table is a list of the proxy temperature data sources used in the 3,000,000 year


reconstruction. Location Type Lat Long Elev. Start

Year 1End

Year 2Citation


Dome Fuji Station D18O -77.32 38.7 3810 358950 50 (Uemura, et al., 2012)

NA87-22/NA87-25/ODP980 Foram 55.39 -14.72 -2220 429500 500 (Waelbroeck, et al., 2002)

V19-30/ODP677 Foram -1.08 -83.54 -3276 429500 500 (Waelbroeck, et al., 2002)

MD94-101 Foram -42.5 79.42 -2920 429380 5000 (Waelbroeck, et al., 2002)

Atlantic Ocean alkenone -31.47 15.32 -1372 1517112 497630 (McClymont, et al., 2005)

MD06-3018/ Coral Sea Foram -23 166.15 -2470 1543500 0 (Russon, et al., 2010)

ODP806B alkenone 0.32 159.36 -2520 1485000 515000 (McClymont, et al., 2005)

Ontong Java Plateau Mg/Ca 0.32 159.36 -2520 3087400 2336900 (Medina-Elizalde & Lea,

2010)

ODP 722 / Arabian Sea alkenone 16.62 59.80 -2022 3330480 7110 (Herbert, et al., 2010)

ODP 662 / Atlantic Ocean alkenone -1.38 -11.73 -3824 2000000 1356000 (Cleaveland & Herbert,

2007)

ODP 882 / Pacific Ocean alkenone 50.35 167.58 -3244 3638880 4270 (Martínez-Garcia, et al.,

2010)

ODP 1090 / Atlantic Ocean alkenone -42.91 8.9 -3702 3642410 0 (Martínez-Garcia, et al.,

2010)

Table F.2: This is a table of the coefficients obtained using Equation 3 to fit the proxy




1 -0.0122 -41.4 -0.0380 102 -0.327 19.2 0.000258

2 0.0356 0.183 0.0772 -153 -8.82 0.712 0.000501

3 0.367 -7.97 0.765 -62.7 -0.0000596 105000 0.252

4 -0.259 -2.31 0.156 -495 5.78 1.09 0.0298

5 0.00338 45.7 -0.369 -16.1 -1.11 5.65 0.00120

6 0.0144 -6.99 -0.397 4.43 2.96 2.12 0.00362

7 -2.56 -20.5 -0.570 -1.70 0.00000197 3194000 1.38

8 -0.324 1.66 -0.0440 0.187 1.40 4.48 0.0142

Table F.3: This is a table of the coefficients obtained using Equation 3 to fit the smoothed proxy




1 2.32 -0.0354 1.94 2.17 0.0000293 108000 1.62

2 -1.79 1.85 -1.15 1.64 0.0000588 107000 1.57

3 -1.96 -0.551 1.45 0.0644 -0.00000133 4720000 1.34

4 0.321 1.80 3.02 0.544 -0.0000244 258000 0.321

121

APPENDIX G

DETAILED METHODOLOGY AND MATHEMATICA FILE CODE

122

The method used in this paper is based on the works of Dr. Craig Loehle. In his paper “A

2000 Year Global Temperature Reconstruction Based on Non Tree Ring Proxies”, Loehle

describes his method of using interpolation to build a temperature reconstruction using proxy

data. As stated by Loehle, “The pollen based reconstruction of Viau et al. (2006) has data at

100-year interval. Other sites had data at irregular intervals. This data is now interpolated to put

all the data on the same annual basis” (Loehle & McCulloch, 2008). The reasoning was “In

order to use data with non-annual coverage, some type of interpolation is necessary, especially

when the different series do not line up in dating. This interpolation introduces some unknown

error in the reconstruction, but is incapable of falsely generating the major patterns seen in the

results” (Loehle & McCulloch, 2008).

In this paper, linear interpolation was used to put all the data on the same annual basis.

Linear interpolation is the simplest method of getting positions in between the data points. In

this method, the two data points are connected with a straight line. This method was chosen

because it was the simplest reconstruction and has been used by other investigators in the field.

Once the data is interpolated, Loehle states that the, “data in each series was smoothed

with a 29-year running centered mean. This smoothing serves to emphasize long term climate

patterns instead of short term variability” (Loehle & McCulloch, 2008). In this dissertation, the

data was originally smoothed with a 10 year running centered mean. Once the data is smoothed

for each year, “All data was then converted to anomalies by subtracting the mean of each series

from that series. This was done instead of using a standardized date such as 1970 because series

date intervals did not all line up or extend to the same ending date. With a single date over many

decades and dating error, a short interval for determining a zero date for anomaly calculations is

123

not valid” (Loehle & McCulloch, 2008). This dissertation followed the same method and

subtracted the mean of each series from that series. The data was then averaged over all data sets.

As stated by Loehle, “the main significance of the results here is not the details of every wiggle,

which are probably not reliable, but the overall picture of the 2000 year pattern showing the

[Medieval Warm Period] and [the Little Ice Age] timing and curve shape. Future studies need to

acquire more and better data to refine this picture” (Loehle & McCulloch, 2008). As stated in

the work by Loehle, the data was smoothed with a running mean that would “help remove noise

due to dating and temperature estimation error” (Loehle, 2007).

Now, there are advantages and disadvantages to using this linear interpolation method.

The advantage to using this method are that with interpolation, the temperatures between the data

sets are estimated, allowing multiple sets of proxy data to be combined. Also with this method,

the data points in the data sets that were collected stay at the same temperature and time point.

The disadvantage with this method is that the temperature between the data points is estimated

using linear interpolation which on occasion is not the case, especially if the interval of time is

very long.

Linear interpolation is one method of reconstructing the past temperature changes, but it

is by no means the only method that could be used. In the previous method, the temperatures

between each data point was estimated with linear interpolation. Another method that could be

used that doesn’t include interpolation would be to average the data points over equal intervals.

For example, you have four data points in a random time range. Assume that the data points are

at (1.75, 2), (2.5, 3), (3.1, 2), and (5, 1), for example. A way to process the data is to divide the

time range into equal sections, from 0-2, 2-4, and 4-6, then take the points in each zone, average

them and center the result in the zone. In this example, the only point in the 0-2 range would be

124

the point (1.75, 2), so this point would be moved to (1, 2). The points in the 2-4 range are (2.5,

3) and (3.1, 2). These points would be averaged and centered at x=3 so the new point would be

(3, 2.5). The third point would stay at (5, 1). If you use the same time ranges for each data set,

then all the data sets would have the same centered points along the x-axis. This would allow

one to average the different data sets without interpolation since all the temperatures would be on

the same x-axis. The advantage to this method would be that interpolation is not used. There is

no estimation of the data between the given data points. On the other hand, the disadvantage of

this method would be that the data points are moved to a center point in each time range. This

would adjust the data points along the time axis, which will influence the results obtained in

testing for periodic fluctuations. Another disadvantage would be that in this method of data

analysis, the time ranges have to be large enough that the data set with the least data points is the

limiting factor in the resolution of the overall average temperature reconstruction.

Once the temperature data sets has been combined to create a reconstructed temperature

model, it becomes necessary to analyze the model to look for temperature fluctuations with

periodic or quasiperiodic periods. When testing for patterns in data, it is common to test for the

periodic function such as sines or cosines. By using sines and cosines to fit the data sets,

(Scafetta, et al., 2001) and (Ignaccolo, et al., 2004) were able to find periodic behavior in teen

pregnancies. Other papers such as (Loehle and Scafetta) were able to use a periodic function to

best fit temperature changes for the past hundred and fifty years. A period of time of only 50

years may show trends in temperature that fit a sine or cosine pattern. When the time interval is

extended over many years there are intermittent cycles that match simple sines or cosines.

However, these cycles do are monotonic and have the appearance of modulated sine or cosine

patterns. Thus, a new approach was incorporated into this work. In this paper, attempts were

125

made to fit the data to Fourier expansion with only limited success. The better approach was

found to be when modulated sine waves or cosine waves were used. The best fit to the data was

found when a cosine function whose argument was modulated by a cosine function was

employed. The expression used can be seen in Equation 3. This allowed for the best fit function

to be quasi-periodic that provided a very good fit to a significant number of the data sets over

very long time intervals. All of the data were analyzed using the function represented in

Equation 3. The results show a superior fit over all of the previously proposed models

represented in the published literature.

In this work, the temperature reconstruction for multiple regions that was used in

this dissertation was collected from the National Oceanic and Atmospheric Administration

(NOAA) Paleoclimatology Data website. (National Oceanic and Atmospheric Administration,

n.d.)

The data were then analyzed using Mathematica wherein the average of the data sets was

calculated. The average is then subtracted from the data set to set all the data sets around zero.

Before this, some data sets were located around the tropics, while others were located around the

poles. Comparing these data sets does not give a full understanding of the temperature changes

between them. So, by subtracting the average of the temperature for the data set, the different

data sets are referenced to the same level, namely around the 0°C Δ in temperature mark, with

temperatures larger than 0°C Δ in temperature being above average for that location and

temperatures smaller than the average as below average for that location. Once this data is

centered on 0°C Δ in temperature, linear interpolation was used to fill in the interval between

some data points. This was done using Equation 2. There is some error associated with each data

set, due to instrumental reading errors, human errors, etc. The errors are along both the

126

temperature axis and along the time axis. Since a majority of the sources did not provide their

error, it was determined that by averaging multiple different data sets, the errors in the time axis

would average out. The temperature errors were calculated using the standard error of the mean,

discussion of which can be found in Section 1 of Appendix G.

By using this equation, where and are two data points in the temperature

reconstruction, the interval between data points can be reconstructed. Although the temperature

may not change linearly between points, this procedure was chosen to “smooth” some of the data

sets. Once each of the data sets was interpolated, they were averaged together to get a general

trend for temperature change.

Since the Earth experiences multiple periodic fluctuations that can affect its climate,

either in a significant or a minor way, it seemed reasonable to choose a “sinusoidal” function to

analyze the data sets for periodic/quasi-periodic changes. The changing gravitational force on the

Earth’s orbit eccentricity as the planets move in their complex pattern will produce a modulation

of the periodicity of the Earth’s motion about the Sun. Thus, a more precise model requires a

modulated sinusoidal pattern. A sinusoidal function with frequency modulation was chosen to

model the temperature changes. The equation used is given by Equation 3.

Both the Standard Error of the Mean SEM and the Test were determined in this work.

Section 1: Standard Error of the Mean

Standard techniques were used to calculate the standard error of the mean. Initially, it

was necessary to calculate the standard deviation (σ) of the data sets. This can be determined by

using Equation G-1.

127

σ

G-1

Where x is the data point, µ is the mean of the data points, and n is the sample size.

Then to get the standard error of the mean (SEM), Equation G-2 was used. Equation G-2 is

given by:

G-2

Section 2: Test

Pearson’s chi-squared test was utilized in analyzing the data sets. This statistical test was

used to evaluate how likely the observed data points differ from the expected data points, or

theoretical, data. In this equation, the observed data are represented by , and the expected or

theoretical data are represented by . N is the total number of observations made. More detailed

analysis of this procedure are given by Bevington and Robinson in Chapter 4.4 (Bevington &

Robinson, 2003). The Test is obtained by using Equation G-3. Equation G-3 is given by:

G-3

All of the graphs in this work were plotted by using Mathematica and taking into

consideration the data spread predicted by the standard techniques provided above. There are

128

uncertainties in both time and temperature in the data sets. In that two variables are considered

in the data sets the statistical behavior of both of these was calculated to determine what

contribution is made in uncertainty for both of these variables.

Section 3: Mathematica File Code

A summary of the Mathematica File Code that was used to process the data sets is given

below for continuity.

Clear["Global`*"]

StartSessionTime=SessionTime[];

(* Steps: (1.) Import Data, (2.) Graph Complete Data, (3.) Graph Limited Data, (4.) Data No

Null, (5.) Take the Average of the Temperature, (6.) Subtracting the Average from the Main

Data, (7.) Seperating the Good Data from the Bad Data, (8.) Normalizing Good Data, (9.)

Interpolation of the Data, (10.) Aligning Interpolated Data,(11.) Taking the -99 out of the data

set, (12.) Averaged Plots, (13.)IntPlots, (14.) Aligned Plots, (15.) Putting the IDA into one Table

in a CSV, (16.) taking out more -99 to get usable data, (17.) IDA Means, (18.) Error List Plot,

(19.) Combinging the Data into One Graph (20.) Maping the Locations and Exporting the Data

*)

stepscomplete=0;

stepstodo=20;

(*-----------------------------------------------------*)

(* STATUS OF THE NOTEBOOK *)

129

(*-----------------------------------------------------*)

Dynamic[stepsdone,UpdateInterval->0.01];

Dynamic[stepsnotdone,UpdateInterval->0.01];

stepsdone:=stepscomplete;

stepsnotdone:=stepstodo;

smallstepscomplete=0;

smallstepstodo=1;

Dynamic[smallstepsdone,UpdateInterval->0.01];

Dynamic[smallstepsnotdone,UpdateInterval->0.01];

smallstepsdone:=smallstepscomplete;

smallstepsnotdone:=smallstepstodo;

statuschart=Dynamic[PieChart[{{smallstepsdone,smallstepsnotdone},{stepsdone,stepsnotdone}}

,ChartStyle->{{Green,Red},{Green,Red}}],UpdateInterval->0.2];

Refresh[statuschart,UpdateInterval->0.03]

(*-----------------------------------------------------------*)

(*VARIABLES TO CHANGE IN NOTEBOOK *)

(*-----------------------------------------------------------*)

StartTime=-2000;

EndTime=2000;

OTime=EndTime-StartTime;

(* LIMITED GRAPHS *)

StartYear =StartTime;

EndYear =EndTime;

130

(* AVERAGEING *)

AverageStart=StartTime;

AverageStop=EndTime;

(* INTERPOLATED DATA ALLIGNED*)

MeanStartYear=StartTime;

MeanEndYear=EndTime;

YearGridStepSize=100;

MOE=0.001;

CombinedData="Interpolated_Combined_Data"<>DateString[{"Month","_","Day","_","Year"}]

<>"_"<>ToString[OTime]<>".csv";AllAlignedGraphName="Combined_Graph"<>DateString[{

"Month","_","Day","_","Year"}]<>"_"<>ToString[OTime]<>".jpg";

OpacityGraphName="Opacity_Graph"<>DateString[{"Month","_","Day","_","Year"}]<>"_"<>

ToString[OTime]<>".jpg";

CombinedOpacityGraphName="Combined_Opacity_Graph"<>DateString[{"Month","_","Day",

"_","Year"}]<>"_"<>ToString[OTime]<>".jpg";

SmallMapName="Mapped_Locations"<>DateString[{"Month","_","Day","_","Year"}]<>"_"<>

ToString[OTime]<>".jpg";

(*---------------------------------------------------------------*)

(* IMPORTING THE DATA FILES *)

(*---------------------------------------------------------------*)

(* USE WITH JAMES OTTO'S LAPTOP *)

Dir="(/Add Directory here)";

131

Sources=Import[Dir<>"File_Name.csv"];

NumSources=Length[Sources];

(*-------------------------------------------------------------------------------*)

(* LABELING THE COLUMNS IN THE MASTER FILE *)

(*-------------------------------------------------------------------------------*)

filenamecol=1;titlecol=2;yearcol=3;tempcol=4;startrow=5;Continentcol=8;YBPStart=14;

YBPStop=15;

(*-----------------------------------------------------*)

(* IMPORTING THE DATA FILES *)

(*-----------------------------------------------------*)


smallstepstodo=NumSources;

StartTime=SessionTime[];

Data=Table[smallstepscomplete+=1;smallstepstodo-

=1;Import[Dir<>Sources[[dv,filenamecol]]<>".csv"],{dv,1,Length[Sources]}];

stepscomplete+=1;

stepstodo-=1;

EndTime=SessionTime[];

ImportTime = EndTime-StartTime

(*This will List all the Data that was collected, so if there is a problem importing the data, you

can see what data is having the error *)

(* Do[Data[[i,1]];Print[i],{i,1,Length[Data]}] *)

(* To see what the name of the source that is the problem *)

132

(* Sources[[1]];*)

(*-----------------------------------------------------*)

(* COMBINES THE DATA INTO A TABLE *)

(*-----------------------------------------------------*)

DataS=Table[(*Print[dv];*)Data[[dv,Sources[[dv,startrow]];;Length[Data[[dv]]],{Sources[[dv,y

earcol]],Sources[[dv,tempcol]]}]],{dv,1,NumSources}];

(*-----------------------------------------------------*)

(* SETTING UP THE PARAMETERS OF THE GRAPHS *)

(*-----------------------------------------------------*)

Ymin=Table[Min[DataS[[dv,;;,2]]],{dv,1,NumSources}];

Ymax=Table[Max[DataS[[dv,;;,2]]],{dv,1,NumSources}];

Xmin=Table[Min[DataS[[dv,;;,1]]],{dv,1,NumSources}];

Xmax=Table[Max[DataS[[dv,;;,1]]],{dv,1,NumSources}];

SetOptions[ListLinePlot,PlotStyle->{Thickness[0.008]},Frame->True,ImageSize-

>Medium,FrameLabel->{"Years (BCE)","Temperature (C)"},BaseStyle->{FontFamily-

>"Times",FontSize->18}];

(*-------------------------------------------------------------*)

(* THE COMPLETE GRAPHS *)

(*-------------------------------------------------------------*)




CompletePlots=Table[smallstepscomplete+=1;

133

smallstepstodo-=1;ListLinePlot[DataS[[dv]],PlotLabel-

>Sources[[dv,titlecol]]],{dv,1,NumSources}];

stepscomplete+=1;

stepstodo-=1;


CGraphTime = EndTime-StartTime

(*----------------------------------------------------------*)

(* THE LIMITED GRAPHS *)

(*----------------------------------------------------------*)




SetOptions[ListLinePlot,PlotStyle->{Thickness[0.008]},Frame->True,ImageSize-

>Medium,FrameLabel->{"Years (BCE)","Temperature (C)"},BaseStyle->{FontFamily-

>"Times",FontSize->18}];

LimitedPlots=Table[smallstepscomplete+=1;

smallstepstodo-=1;ListLinePlot[DataS[[dv]],PlotRange->{{StartYear,

EndYear},Full},PlotLabel->Sources[[dv,titlecol]]],{dv,1,NumSources}];

stepscomplete+=1;

stepstodo-=1;


LGraphTime = EndTime-StartTime

(*------------------------------------------------------------*)

134

(* TAKING THE AVERAGE *)

(*------------------------------------------------------------*)




DataNoNull=Table[smallstepscomplete+=1;

smallstepstodo-=1;Select[DataS[[dv]],#[[2]]!=""&&#[[2]]>-272.3&],{dv,1,NumSources}]//N;

(* There must be at least (the following) number of data points in the selected year range (above)

to take the average otherwise warn me *)

AverageMinPoints=10;

(* sources with too few will have their source number stored *)

LowPoints={};

NoPoints={};

stepscomplete+=1;

stepstodo-=1;


DataNoNullTime = EndTime-StartTime




Average=Table[smallstepscomplete+=1;

smallstepstodo-=1;

Clear[tempVar];

135

tempVar=Select[DataNoNull[[dv]],#[[1]]>=AverageStart&&#[[1]]<=AverageStop&];

Which[

Length[tempVar]>=AverageMinPoints,

Mean[tempVar[[;;,2]]],

Length[tempVar]>0&&Length[tempVar]<10,

AppendTo[LowPoints,dv];

Print["Warning too few points in source "<>ToString[dv]];

Mean[tempVar[[;;,2]]],

Length[tempVar]<1,

AppendTo[NoPoints,dv];

Print["Warning NO points in source "<>ToString[dv]];

-99

],{dv,1,NumSources}];

stepscomplete+=1;

stepstodo-=1;


TakingAverageTime = EndTime-StartTime

(*------------------------------------------------------------------*)

(* SUBTRACTING THE AVERAGE *)

(*------------------------------------------------------------------*)

CompleteSources=DataNoNull[[Complement[Table[dv,{dv,1,NumSources}],NoPoints]]];

NumCS=Length[CompleteSources];

136

(* YearGridStepSize=Ceiling[Mean[Flatten[Table[Table[Abs[DataNoNull[[dva,dvb+1,1]]-

DataNoNull[[dva,dvb,1]]],{dvb,1,Length[DataNoNull[[dva]]]-1}],{dva,1,NumCS}]]]/2];*)




NormalizedData=Simplify[Table[

Table[smallstepscomplete+=1;

smallstepstodo-=1;{CompleteSources[[dv,dv1, 1]],CompleteSources[[dv,dv1,2]]-

Average[[dv]]},{dv1,1,Length[CompleteSources[[dv]]]}]

,{dv,1,NumCS}]//N];

stepscomplete+=1;

stepstodo-=1;


SubtractingAverageTime = EndTime-StartTime

(*-------------------------------------------------------------------------------------*)

(* SEPERATING GOOD DATA SETS FROM BAD DATA SETS *)

(*--------------------------------------------------------------------------------------*)




(* The variable messedup (below) will contain a single entry for each source with duplicates.

The entry is {Number of dublicates, the number of the source} --- You can check the data by

looking at NormalizedData[[number of the source with errors]]*)

137

Length[messedup]

Length[NormalizedData]

NumSources

messedup=DeleteCases[Table[smallstepscomplete+=1;

smallstepstodo-

=1;CDist=CountDistinct[NormalizedData[[dv,;;,1]]];NumEnts=Length[NormalizedData[[dv,;;]]]

;If[CDist<NumEnts,{NumEnts-CDist,dv},-99],{dv,1,NumCS}],-99];

MessedUpSources=messedup[[;;,2]];

stepscomplete+=1;

stepstodo-=1;


SeperatingTime = EndTime-StartTime

(* Check Graphs for Problems*)

LimitedAveragedPlots=Table[ListLinePlot[NormalizedGoodData[[dv]],PlotRange-

>{{StartYear, EndYear},Full},PlotLabel->Sources[[dv,titlecol]]],{dv,1,NumSources}];

(*------------------------------------------------------------------------*)

(* MAKING A TABLE OF GOOD SOURCES *)

(*------------------------------------------------------------------------*)




MessedUpSources

GoodSources=Table[dv,{dv,1,NumCS}];

138

GoodSourcesTemp=GoodSources;

Do[GoodSourcesTemp=Cases[GoodSourcesTemp,Except[MessedUpSources[[dv]]]],{dv,1,Leng

th[MessedUpSources]}]

GoodSources=GoodSourcesTemp

NumGoodSources=Length[GoodSources];

GoodFraction=NumGoodSources/NumSources//N

NormalizedGoodData=Table[smallstepscomplete+=1;

smallstepstodo-=1;NormalizedData[[GoodSources[[dv]]]],{dv,1,NumGoodSources}];

stepscomplete+=1;

stepstodo-=1;


GoodSourcesTime = EndTime-StartTime

(*---------------------------------------------------------------*)

(* INTERPOLATING THE DATA *)

(*---------------------------------------------------------------*)




(* Interpolates each set of Normailized Good Data *)

IntFunctions=Table[smallstepscomplete+=1;

smallstepstodo-=1;Interpolation[NormalizedGoodData[[dv]],InterpolationOrder-

>1],{dv,1,NumGoodSources}];

stepscomplete+=1;

139

stepstodo-=1;


IntFunctionsTime = EndTime-StartTime




Years=Table[dv,{dv,MeanStartYear,MeanEndYear,YearGridStepSize}];

numYears=Length[Years];

IntData= Table[smallstepscomplete+=1;

smallstepstodo-=1;

MinY=Min[NormalizedGoodData[[dv,;;,1]]];

MaxY=Max[NormalizedGoodData[[dv,;;,1]]];

Table[

dvy=Years[[dv2]];

If[

dvy>MinY&&dvy<MaxY,

{dvy,IntFunctions[[dv]][dvy]}

,

{dvy,-99}

]

,{dv2,1,numYears}]

, {dv,1,NumGoodSources}];

140

IntDataAligned =IntData;

stepscomplete+=1;

stepstodo-=1;


IntDataTime = EndTime-StartTime

(*------------------------------------------------------------------*)

(* GRAPHS OF AVERAGED DATA *)

(*------------------------------------------------------------------*)




IDAShort=Table[smallstepscomplete+=1;

smallstepstodo-=1;DeleteCases[IntDataAligned[[dv]],x_/;x[[2]]==-

99],{dv,1,NumGoodSources}];

stepscomplete+=1;

stepstodo-=1;


IDAShortTime = EndTime-StartTime




AveragedPlots=Table[smallstepscomplete+=1;

141

smallstepstodo-=1;ListLinePlot[IDAShort[[dv]],PlotRange->{{StartYear,

EndYear},Full},PlotLabel->Sources[[dv,titlecol]]],{dv,1,NumGoodSources}];

stepscomplete+=1;

stepstodo-=1;


AveragedPlotsTime = EndTime-StartTime

(*--------------------------------------------------------------*)

(* COMBINING THE GRAPHS *)

(*--------------------------------------------------------------*)




SetOptions[ListPlot,PlotStyle->{Thickness[0.008]},Frame->True,ImageSize-

>Large,FrameLabel->{"(BCE) Years (CE)","Temperature

(°C)"},BaseStyle->{FontFamily->"Times",FontSize->18}];

IntPlots=Show[Table[smallstepscomplete+=1;

smallstepstodo-=1;ListPlot[IntData[[dv]],PlotStyle-

>Black],{dv,1,NumGoodSources}],PlotRange->{{MeanStartYear,MeanEndYear},{-7,7}}]

stepscomplete+=1;

stepstodo-=1;


IntPlotsTime = EndTime-StartTime


142



AlignedPlots=Show[Table[smallstepscomplete+=1;

smallstepstodo-=1;ListPlot[IntDataAligned[[dv]],PlotStyle-

>Black],{dv,1,NumGoodSources}],PlotRange->{{MeanStartYear,MeanEndYear},{-5,5}}]

stepscomplete+=1;

stepstodo-=1;


AlignedPlotsTime = EndTime-StartTime




(* Putting all the data plots into one table starting from the Start Year to the End Year *)(* add

small timer to rest *)

IDACSV=Table[smallstepscomplete+=1;

smallstepstodo-

=1;Join[{IntDataAligned[[1,dvy,1]]},IntDataAligned[[;;,dvy,2]]],{dvy,1,Length[Years]}];

stepscomplete+=1;

stepstodo-=1;


IDACSVTime = EndTime-StartTime



143


(* Taking out the -99s to only get the usable data *)

YearsDataPts=Table[smallstepscomplete+=1;

smallstepstodo-=1;DeleteCases[IDACSV[[dvy,2;;NumGoodSources+1]],-

99],{dvy,1,Length[Years]}];

stepscomplete+=1;

stepstodo-=1;


YearDataTime = EndTime-StartTime




(* Master Table with the Year, columns of data points, Mean, Standard Deviation, and # of data

points used to get the mean *)

IDAMeans=Table[smallstepscomplete+=1;

smallstepstodo-

=1;Join[IDACSV[[dvy,;;]],{Mean[YearsDataPts[[dvy]]],StandardDeviation[YearsDataPts[[dvy]

]],Length[YearsDataPts[[dvy]]]}],{dvy,1,Length[Years]}];

stepscomplete+=1;

stepstodo-=1;


IDAMeansTime = EndTime-StartTime

Needs["ErrorBarPlots`"]

144




SetOptions[ErrorListPlot,PlotStyle->{Thickness[0.008]},Frame->True,ImageSize-

>Large,FrameLabel->{"(BCE) Years (CE)","Temperature

(°C)"},BaseStyle->{FontFamily->"Times",FontSize->18}];

ELP=ErrorListPlot[Table[smallstepscomplete+=1;

smallstepstodo-

=1;{IDAMeans[[dvy,1]],IDAMeans[[dvy,NumGoodSources+1+1]],IDAMeans[[dvy,NumGood

Sources+1+2]]},{dvy,1,Length[Years]}],PlotRange->{{MeanStartYear,MeanEndYear},{-

4,2}},PlotLabel->"Graph of Average Data with Error Bars"]

(* NEED TO FIX THE RANGE OF THE PLOT*)

SetOptions[ListPlot,PlotStyle->{Thickness[0.008]},Frame->True,ImageSize-

>Medium,FrameLabel->{"(BCE) Years (CE)","Number of

Sources"},BaseStyle->{FontFamily->"Times",FontSize->18},PlotLabel->"Number of Sources

Used in 14000 Year Temperature Reconstruction"];

ListPlot[IDAMeans[[;;,{1,NumGoodSources+1+3}]]]

Dir="/Users/James/Dropbox/Research/ExportedData/";

CreateDirectory[Dir<>DateString[{"Month","_","Day","_","Year"}]];

Export[Dir<>DateString[{"Month","_","Day","_","Year"}]<>"/"<>CombinedData,IDAMeans];

stepscomplete+=1;

stepstodo-=1;


145

ErrorPlotTime = EndTime-StartTime

(*----------------------------------------------------------------------*)

(* PLOTTING THE COMBINED GRAPHS *)

(*----------------------------------------------------------------------*)




AllTogether=Table[{IDAMeans[[dvy,1]],IDAMeans[[dvy,NumGoodSources+1+1]]},{dvy,1,Le

ngth[Years]}];

SetOptions[ListPlot,PlotStyle->{Thickness[0.008]},PlotLabel->"14000 Years of Temperature

Reconstructions",Frame->True,ImageSize->Large,FrameLabel->{"(BCE) Years

(CE)","Temperature (°C)"},BaseStyle->{FontFamily->"Times",FontSize->18}];

ATG=AllTogetherGraph=ListPlot[AllTogether,Joined->True,PlotStyle->Black]

OpacityGraph=Show[Table[smallstepscomplete+=1;

smallstepstodo-=1;ListPlot[IntData[[dv]],PlotStyle-

>Opacity[0.08]],{dv,1,NumGoodSources}]];

CombinedOpacityGraph=Show[OpacityGraph,AllTogetherGraph,PlotRange-

>{{MeanStartYear,MeanEndYear},{-2,2}},PlotLabel->"Graph of Combined Data Sets

Overlayed with Average Data",Frame->True,ImageSize->Large,FrameLabel->{"(BCE)

Years (CE)","Temperature (°C)"},BaseStyle->{FontFamily->"Times",FontSize-

>18}]

Show[{ELP,ATG},PlotRange->{{-12000,2000},{-4,2}},Frame->True,ImageSize-

>Large,PlotLabel->Style["Average Temperature Function with Error Bars",FontSize-> 20,

146

Bold],FrameLabel->{"(BCE) Years (CE)","Temperature

(°C)"},BaseStyle->{FontFamily->"Times",FontSize->18}]

stepscomplete+=1;

stepstodo-=1;


LastGraphTime = EndTime-StartTime

(*--------------------------------------------*)

(* Graphing the Locations *)

(*--------------------------------------------*)




Locations=Sources[[1;;Length[Sources],{9,10}]];

SmallLocationMap=Graphics[{

Gray,Polygon[Map[GeoGridPosition[GeoPosition[#],"Mercator"][[1]]&,CountryData[#,"Coordi

nates"],{2}]]&/@CountryData["Countries"],

Red,PointSize[.006],Point/@Map[GeoGridPosition[GeoPosition[#],"Mercator"][[1]]&,{Locatio

ns},{2}]}]

Dir="C:/Users/Owner/Desktop/";

CreateDirectory[Dir<>DateString[{"Month","_","Day","_","Year"}]];

147

Export["C:/Users/Owner/Desktop/Research/Graphs/"<>DateString[{"Month","_","Day","_","Ye

ar"}]<>"/"<>AllAlignedGraphName,AllTogetherGraph];


ar"}]<>"/"<>OpacityGraphName,OpacityGraph];


ar"}]<>"/"<>CombinedOpacityGraphName,CombinedOpacityGraph];


ar"}]<>"/"<>SmallMapName,SmallLocationMap];

stepscomplete+=1;

stepstodo-=1;


MapTime = EndTime-StartTime


EndTime-StartTime (* How Much Time Program took to Run *)

EndSessionTime=SessionTime[];

OverallRunTime=EndSessionTime-StartSessionTime

148

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