LAB 1
GRAPHING CLIMATOLOGICAL AND METEOROLOGICAL DATA
Objectives:
-To become familiar with a few of the many types of relations that can be illustrated graphically.
-To become familiar with the terminology used in analyzing a graph.
-To create and analyze graphs of meteorological data.
Numerical data are often represented by a graph. Graphs provide a quick method for qualitatively and quantitatively analyzing the relation between 2 (or more) variables. Graphs are used extensively in many fields because they often are the most efficient way to summarize how variables relate to one another. In G107, you will see many graphs that are used to illustrate different phenomena: variation of air temperature at various time scales, change of atmospheric pressure with altitude, velocity of flow in rivers, etc.
I. GRAPHING
1. IDENTIFICATION OF INDEPENDENT AND DEPENDENT VARIABLES AND ORIENTATION OF THE GRAPH
A graph can be created when one has simultaneous observations for 2 or more variables. For example, suppose you record the ambient air temperature and time of observation every hour for 48 hours. You could create a graph, which shows how the temperature varied with time since you know both the temperature and the time for every observation you made. If you had merely recorded the temperature and not the time, no such graph would be possible as you would only have observations for one variable. By convention a graph consists of the following:
The axes consist of 2 perpendicular lines: one horizontal, one vertical, joined at the origin (the value of data at the origin is not always zero).
The horizontal axis is the x-axis and is associated with the independent variable (the variable which is thought to influence the other variable).
The vertical axis is the y-axis and is associated with the dependent variable (the variable which is influenced by the independent variable). To identify the dependent variable, you should ask yourself the following question: Which one is the most likely to change with the other one?
Continuing the example from above, we would choose time as the independent variable and plot it along the x-axis since it is much more likely that time influences air temperature than air temperature influences time.
2. SCALING THE AXIS
Axes must be scaled to ensure that each data point is placed correctly in relation to all other points and all of the points fit on the graph. Suggested steps in order to scale axes:
1) RANGE: Determine the smallest and largest value to be plotted for each of the variables; i.e., the range to be plotted on each axis. You do not have to start at zero.
2) SPACE AVAILABLE: Determine the available space on the graph paper; i.e., the number of major subdivisions shown along each of the axes of the graph paper
3) SCALE THE AXIS: Divide RANGE by the SPACE AVAILABLE and round the value to a convenient increment (1, 2, 5, 10 etc.). This will give you a constant increment for each of the two axes, which will allow the range of data to fit on the page, filling as much of the page as possible. Usually, we need to round up the value of the increment in order to have enough space to plot the data on the axis. Remember each axis does not have to start at zero.
NOTE: you do not have to use the same scaling on both axes.
EXAMPLE:
In an air temperature series we have the following values:
Minimum temperature = -20° C
Maximum temperature = 31° C
RANGE: 31° – (-20°) = 51°
SPACE AVAILABLE: 18 boxes
SCALE THE AXIS: 51° /18 boxes = 2.83° per box (this means that 1 box is equal to 2.83° )
You should round up to 3° per box to make sure you will have enough space to plot all the data on the graph. If you round down, you will run out of space to plot all your data on the graph.
Axes commonly have two types of scales:
a) Linear: equidistant divisions
b) Logarithmic: major divisions represent multiplication by a constant factor (most commonly 10) rather than addition as in the linear case.
NOTE: there is no zero on this scale.
Logarithmic scales are very useful if data across a very wide range is to be plotted.
For more discussion on the logarithmic scale, see section below on exponential relations. We will use this type of scale in Lab 5.
3. COMPLETING THE GRAPH
Titles and labels
Title: The graph should have a title, which conveys information concerning the content of the graph (what is the graph representing exactly).
Source: Where relevant, the source of the data should be indicated.
Axes: Each axis should be labeled to indicate the variable being graphed. The units must be included. Without units, data have no meaning.
Data points (to connect or not to connect!)
Each observation in the data set is plotted as a point on the graph. To do this, locate the intersection of a vertical line extended from its value on the x-axis; and a horizontal line extended from its value on the y-axis. Mark the data point either with a small cross or a dot with a circle around it, so that the position of the data point can be seen accurately. The resulting plot of all the points is known as a scatter plot. If the data represent a series collected through time the points may be joined in sequence to produce a time series (e.g. temperature through the day). If the data represent a series collected through space, the points will also be connected (e.g. change of temperature with altitude).
Time series data are commonly presented as two kinds of graphs: line graphs and bar graphs. Line graphs should only be used where each point on the graph relates to the magnitude of the variable under consideration at a point in time, e.g. graphs of temperature, relative humidity, wind speed, solar radiation, etc. (these are continuous data). Where magnitudes are related to periods of time, e.g. daily rainfall, monthly snowfall etc., bar graphs are more appropriate (these are referred to as discrete data).
When there is a time series where the data are taken at a point in time (or space), but are not collected at regular intervals (i.e., not every hour, every km, or every 5 years), you cannot connect the data points with a continuous line: you should use a dotted line. We will see an example of that in Lab 5.
Where there are reasonable grounds for believing that magnitudes have changed steadily, but not necessarily in a linear fashion, between observations (as with temperature collected every day), a smooth curve may be appropriate. When considerable fluctuations occur between observations, points should be joined by straight lines rather than by curves.
II. TYPES OF RELATIONS
If the data are not a time or space series, but reflect observations collected to look at the relationship between two variables, the scatter plot can be analyzed in different ways. Visual examination of scatter plots allows a qualitative assessment of the relation between the two variables. A best-fit line may be drawn through the data to illustrate this relation. In this specific case, the data points will not be connected.
If the data fall on a straight line, the line may be drawn by joining adjacent points with a ruler.
If the data do not fall on a straight line, then it is necessary to determine where the line should be drawn. The location of this line may be determined mathematically (regression line) or by "eye". The by "eye" method requires a line to be drawn through the middle of the data such that the scatter of points is minimized around the line.
There may be no apparent relation or the relation may exhibit some recognizable characteristics. While there are many types of relations, which could be illustrated with scatter plots, we will concentrate on only a few, which occur frequently in physical geography: linear, periodic, and exponential.
1. Linear relations are those in which the data points fall on a straight line.
A straight line can be described mathematically by the following equation:
Where: y and x refer to data point values on the y and x-axis, respectively;
m is the slope of the line (the change in y for a unit change in x: the rise/run);
b is the intercept (the value of y when x = 0).
Given any 2 points on the line, the slope (m) and the intercept (b) may be determined as follows:
Where (x1, y1) and (x2, y2) are the x and y values of the 2 points (see diagram above)
NOTE: In further labs, we will also use the term gradient to refer to the slope. They are synonyms.
Alternatively, the intercept can be determined graphically by reading off the intercept when x=0 (see diagram above). Note that eq. 1.2 is also the equation for a gradient, the change in a variable over a distance. We will return to this concept in Labs 3 and 5.
A linear relation is characterized as positive if its slope is greater than 0 (the line points up towards the right). The relation is negative if its slope is less than 0 (the line points down toward the right).
The distinguishing feature of a linear relation is that a change in the independent variable produces the same change in the dependent variable regardless of the actual values of the independent variable. In other words, the rate of change in y for a change in x (i.e., the slope) is constant. Hence, for a linear relation the slope can be calculated between any two data points. Despite this fact, it is always good to select data points that are far apart from each other in order to calculate the slope accurately.
2. Periodic relations are non-linear relations in which the variation of the values of y forms a pattern, which is repeated for all values of x. The sine and cosine functions in trigonometry are examples of periodic relations. Periodic relations occur frequently in meteorology and climatology because many of the variables of interest (temperature, humidity, pressure, radiation, etc.) have daily and/or annual cycles.
3. Exponential relation.
This relation differs from a linear relation in that, for a positive relation, the dependent variable (y) increases at an increasing rate with increases in the independent variable (x). For a negative exponential relation, the dependent variable decreases at a decreasing rate with increases in the independent variable.There are many examples of exponential relations in physical geography. One of the first you will come across is the variation of pressure with height in the atmosphere (see first graph below).
EXAMPLE:
If we calculate the gradient of the change of pressure (dependent variable) with altitude (independent variable) at two different locations on the curve, we will understand better the exponential relation.
From 0 to 5000 m, the pressure changes from 1013 mb to 550 mb.
m = 550 mb – 1013 mb = -0.09 mbm-1
5000 m – 0 m
From 5000 m to 10,000 m, the pressure changes from 550 mb to 250 mb.
m = 250 mb – 550 mb = -0.06 mbm-1
10,000 m – 5000 m
We thus see that for this case, which is a negative exponential relation, the dependent variable (atmospheric pressure) decreases at a decreasing rate with increases in the independent variable (altitude).
Exponential relations may be transformed to linear relations by plotting the logarithm of the dependent variable against the independent variable. This can be achieved by either taking the logarithm of the variables and plotting them, or plotting the values on logarithmic graph paper (see second graph below).
