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\centerline{MONT 100N -- Modeling the Environment}
\centerline{Chapter 4 Project -- The Mauna Loa Atmospheric CO2 Data Set.}
\centerline{October 18, 2017}
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\noindent
The Mauna Loa Observatory (located at an elevation of about 3400 meters on the Mauna Loa volcanic 
mountain on the ``big island'' of  Hawaii) is a research station maintained and staffed by the 
National Oceanic and Atmospheric Administration (NOAA), the major agency of the US government that 
studies weather and climate phenomena.   NOAA maintains a web site, for instance, with the National 
Weather Service day-to-day forecasts and severe weather warnings that form the basis for most local 
weather reporting in the media.   Among the data collected regularly at  Mauna Loa are measurements 
of atmospheric concentrations of a host of trace gasses, including carbon dioxide, or CO2.  
The data set of those measurements goes back to 1958 and is one of the most complete records of 
the recent evolution of this aspect of the Earth's atmosphere.   In this project we will apply the 
modeling techniques introduced in this chapter to try to understand what this data set is saying 
about changes in atmospheric CO2 over time. 
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\noindent
{\it Important Note}
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This is a well-known data set and you can find all sorts of discussions of various aspects of 
it on the web, if you look.  I am going to ask all of you \emph{not to look} until after you have 
worked through at least questions A through D below, though.   The idea is for you to approach this 
entirely ``fresh'' and make your own observations and analysis and draw your own conclusions.
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\noindent
{\it Getting Started}
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The data we will be looking at is contained in a (large) Excel spreadsheet called 
{\tt MaunaLoaCO2Data.xls}  that you will download from 
the course homepage.  Begin by getting the spreadsheet and opening it in Excel.  
Note the layout: 
\begin{itemize}
\item Column A gives the year the measurement was taken
\item Column B gives the month (1 = January, through 12 = December)
\item Column C gives a decimal equivalent of the middle of the month, 
so for instance January 1958 is given as 1958.04, since 1 month = 1/12 year = .08 year (roughly), 
and .04 year is about 15 days.
\item Column D gives the average CO2 level observed at Mauna Loa that month in units of parts 
per million
\item In Column D, if you look closely, you will see that a few of the entries near the start 
are -99.99.  What do you suppose that means?  
\item In Column E, you will notice that most of the entries are the same as the corresponding 
entries in Column D, but the -99.99 entries have been replaced by other values.  These are 
``interpolated'' (estimated) values based on the trends from the nearby months.   We will use 
Column E for all our values so that the -99.99's are not included.   
\end{itemize}
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\noindent
{\it Questions}
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The first thing you will notice if you look at the CO2  levels is that there is \emph{a lot} of
up-and-down variation.  Is it completely random, though?  And is there an underlying trend?
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\noindent
(A)  To start to answer this question:
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\begin{enumerate}
\item[(1)] Create scatter plots of the CO2 monthly averages for the calendar years 
1965, 1975, 1985 (individually), versus the decimal year from Column C.   This will 
require picking out the correct range of rows in Columns C and E for each of these years, 
and you may want to copy those values to other cells to create the scatter plots.  \vskip 5pt
\item[(2)] Looking at these scatter plots, what do you notice about the way CO2 levels 
vary over these years?  Describe what happens over the course of a typical year, and 
hypothesize a reason why the annual pattern works this way.  Note:  Mauna Loa is in the 
Northern Hemisphere and typical mixing patterns in the atmosphere mean that most of the 
air that passes over this location has come from other areas in the Northern Hemisphere.   
What happens through the months of May, June, July, August in the Northern Hemisphere, and 
how might that affect atmospheric CO2 levels?  \vskip 5pt
\item[(3)]  \emph{Extra Credit}: How might you model the yearly variation of the 
CO2 readings?  Suggest mathematical function(s) that might be useful and how you might 
apply them.
\end{enumerate}
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\noindent
(B) Condensing the Data to a More Manageable Form. Our goal is to model how atmospheric 
CO2 levels have been changing over this period (but on the year-to-year level, not 
on the much more variable month-to-month level).  This will be much more manageable if we 
identify some way to compute a ``summary value'' for each year to use as the representative 
CO2 level for that year.   \vskip 5pt
\begin{enumerate}
\item[(1)]  Identify (at least) three different ways that might be used to produce that sort 
of ``summary value'' and describe why they would be suitable.  \vskip 5pt
\item[(2)]  Choose one of your proposed ways to do this and give a reason for why you think  
that will be a reasonable way to ``condense'' the data for each year. \vskip 5pt
\item[(3)] Create new columns in your spreadsheet giving the number of years since 1959, and 
your summary CO2 value for the year.    Since we don't have complete values for the years 
1958 or 2017, just use the years 1959 to 2016 (58 years in all).
\end{enumerate}
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\noindent
(C)  ``Let the modeling begin!'' \vskip 5pt
\begin{enumerate}
\item[(1)]	Using Excel, fit a linear model to your ``condensed'' data set and record 
your results.  Give the equation of the regression line as a function of the years since 
1959 (that is, $x = 0$ correspond to 1959, $x = 1$ corresponds to 1960, and so forth).  Also 
give the $R^2$ value reported by Excel as a measure of goodness of the fit, and discuss the 
residuals for the linear model (in particular, is there a consistent pattern there)?  \vskip 5pt
\item[(2)]  What does your model predict concerning the CO2 level in 2020?  (This is slightly 
outside the interval 1959 to 2016 of course, but not too far outside. So extrapolation from 
the linear model is at least a possibility!)
\end{enumerate}
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\noindent
(D) Atmospheric CO2 levels are of concern, of course, because of the ``greenhouse gas'' 
properties of this compound--the way atmospheric CO2 can trap energy from reflected solar 
radiation and increase temperatures near the surface of the Earth.  Some greenhouse effect 
is necessary for life on Earth, of course (our water-based form of life could not exist at 
the temperatures that would prevail with no greenhouse effect at all because all water would 
be frozen as ice).   But have there been times in the past when CO2 concentrations were 
significantly higher than they are now?  What were the Earth's climate and sea levels like 
then?  (This may require some research--be sure to give the sources you used to compile your 
information.)
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\noindent
Assignment:  Submit your edited Excel spreadsheet with the data and write up answers to the questions above
in a separate document.
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