Graphs and Representation

1. Graphs and Representation
Prepared by
Samikshya Acharya (Roll no. 10)
Saurav Poudel (Roll no. 11)
Submitted to:
Dr. Jeevan Kafle
Central Department of Geology
Kirtipur, Kathmandu

2. Contents
 Graphs and graphical representation
 Use of graph in geology
 Log normal and Log - Log graph
 Triangular diagram
 Polar graphs
 Equal Interval, Equal Angle and Equal area Projections of a Sphere:

3. What is Graphs & Graphical Representation??
In field of applied mathematics a graph is
defined as a chart with statistical data
which are represented in the forms of
curves or lines down across the coordinate
point plotted in it’s surface.
A graphical representation refers to the use
of charts and graphs to visually display,
analyze, clarify and interpret numerical
data, functions and other qualitative
structures.
Representational graphics can quickly
illustrate general behavior and highlight
phenomenon, anomalies and relationships
between the data points.

4. Why Engineering Geologist Uses graphs??
In the field of geology predicting
and understanding the physical
way that the earth works.
So the geologist uses graphs in
the various ways:
i. To visualize, predict
and forecasting geologic
events.
ii. To visualize the large
sizes of data and to
study about system
behavior.
iii. To visualize the study
results.

5. Log-normal and log-log graphs
• The use of logarithms, to enable a wide spread of data to be visualized:
For Example: Table below gives the masses of various modern and extinct animals
together with the total areas of the soles of their feet (this is relevant to whether
these animals could walk on soft mud without sinking in and can help to indicate the
environment in which they lived). This data is plotted on an x–y type graph in Fig.
below.

6. Table: Masses and total foot area for modern and
extinct animals. Data taken from Alexander, R.
(1989). Dynamics of Dinosaurs and other Extinct
Giants, Columbia University Press, New York.
Figure: Simple x–y plot of the data in Table.

7. In Table above, the foot areas are also spread over a rather large
range. It might be useful, therefore, to take logarithms of the areas or
to use logarithmically scaled axes in both directions. Figure below is
an example of such a log-log plot.
Incidentally, in this example, using a logarithmic scale for the vertical
axis has only made a marginal improvement. However, in other cases
it will make a much more useful alteration in the distribution of the
data across the graph.

8. Figure: The data from Table plotted using logarithmic
axes in both directions.

9. Triangular diagrams
Triangular diagrams can be used whenever you wish to visualize the relative
proportions of three components making up a specimen. Common examples are:
(i) The proportions of sand (particles between 2 and 0.063 mm diameter), silt
(0.063–0.004 mm) and clay (less than 0.004 mm) in a sedimentary rock;
(ii)An AFM diagram which shows the proportions of alkalis, iron and
magnesium in a volcanic rock.
Figure shows simple cases from the sedimentological example. Point A sits in
the corner marked ‘100% clay’ and represents a sediment containing only clay.
 Similarly, points B and C represent rocks containing exclusively sand and silt,
respectively.

10. • (100% Clay)A
• (100% Clay)A
(100% Clay)A C (100% Silt)
F
D E
G
B (100% Sand)
Figure: A triangular diagram showing the proportion of clay, sand and silt for seven different sedimentary rock
specimens.

11. Point D lies half way along a line joining 100% clay to 100% sand.
It represents a sediment half of which is clay and half of which is
sand. Similarly, point E is a 50 : 50 sand–silt mixture whilst point F is
a 50 : 50 clay–silt mixture.
Finally, point G is in the center of the triangle, equidistant from all
three edges, and represents a sediment which consists of a 1/3 clay, 1/3
sand and 1/3 silt mixture.

12. Use Fig. 6.8 or some triangular graph paper to plot an AFM diagram as follows. The
left corner of the plot represents 100% (Na2O + K2O). The right hand corner
represents 100% MgO.
The top corner represents 100% (FeO + Fe2O3). Mark these points on your graph
and then plot the following data which is taken from a set of related volcanic rocks.
(i) 10% (Na2O + K2O), 45% MgO, 45% (FeO + Fe2O3). (ii) 10% (Na2O + K2O),
35% MgO, 55% (FeO + Fe2O3).
(iii) 10% (Na2O + K2O), 25% MgO, 65% (FeO + Fe2O3). (iv) 12% (Na2O + K2O),
20% MgO, 68% (FeO + Fe2O3).
(v) 15% (Na2O + K2O), 15% MgO, 70% (FeO + Fe2O3). (vi) 18% (Na2O +
K2O), 12% MgO, 70% (FeO + Fe2O3).
(vii) 23% (Na2O + K2O), 12% MgO, 65% (FeO + Fe2O3).
A graph such as this can furnish significant information about the evolution of a
volcanic rock series.

13. Figure: A triangular diagram net. Lines are drawn here at 10% increments
although most such graph paper would also have 1% intervals marked.

14. Polar Graphs
Some types of data are naturally cyclic.
For example, the data in table below gives the strength of the non-dipole portion
of the Earth’s magnetic field at various locations around the equator.
Note: (the non-dipole field is that part of the magnetic field which can’t
be explained as due to a simple bar magnet).
The data at a longitude of 330° is only 30° from the data at 0° but, on an x–y plot,
it would appear at the opposite end of the graph.
Using a polar plot avoids this problem.
In the polar plot, the longitude is plotted around the circumference of a circle
whilst the field strength is given by the distance from the plot centre (i.e. the
stronger the field the further the point is from the centre).

15. • Table: Non-dipole magnetic strength (in micro tesla) at various locations around
the Earth’s equator.
Figure: The data from Table plotted in polar form.

16. Equal Interval, Equal Angle and Equal area Projections of a
Sphere:
This section deals with the problem of plotting data measured on the surface of a
sphere onto a flat sheet of paper.
This cannot be done without distortion and there are therefore a large number of
different ways of doing this.
methods which are widely used in structural geology, crystallography, earthquake
seismology and many other branches of Earth science.
These projections are useful whenever information about directions in three
dimensions is plotted.

17. In this section I introduce
some of these ideas but, it
must be emphasized,
application in particular fields
has much more extensive uses
than those described here.
Is the bed a simple planar
dipping one or is the bed
folded in some way? If the
bed is planar, what is the true
dip and dip direction?

18. This diagram below assumes that the bed is indeed a simple dipping planar bed
represented by the dipping plane in this figure.

19. The sphere as having lines of latitude and longitude in a similar fashion to the
Earth, each point plots at a ‘southerly’ latitude equal to the apparent dip and at a
longitude equal to the azimuth.
Thus, if we plot the projection of our dip measurements onto a sphere and if the
resulting points lie along a great circle, the measured bed is a simple dipping
plane.
On the other hand, if this spherical projection of the dip measurements is not a
great circle, the bed is not planar. Note that, because dips are always measured
below the horizontal only half of all possible orientations are represented.
Thus, the spherical projection of the dip data should actually define a semicircle
(i.e. the lower half of the great circle).

20. Plotting and performing measurements on sphere is not very convenient. A
solution would be to plot the data using projection which represents the surface of
the sphere on a flat sheet of graph paper.
Deal with the lower hemisphere, is to plot onto polar graph paper.
 To do this .the azimuth is plotted around the circumference of the plot and
distance, r, from the centre of the plot is used to represent dip.
In other words, dip increases linearly from zero at the plot circumference to 90° at
the plot centre.
Thus, if a point on the plot represents a dip of φ, the corresponding distance, r,
from the plot centre is given by
 r = R(90-Φ)/90
 where R is plot radius.

21.  For example, a dip of 45° would plot at a distance of
r = R(90 − 45)/90
= R/2
I.e. half way between the plot edge and the plot centre. Similarly, a dip of 0° would
be at the plot edge (i.e. r = R) and a dip of 90° would be at the graph centre (i.e. r =
0).
Using this method for plotting the measurements in Table 6.4 results in Fig. 6.12
and is known as an equal interval projection.

22. Fig. 6.12 Equal interval polar plot of the data
Fig. 6.13 Equal angle (stereographic) projection of the data from Table 6.4. A great
circle is also shown which passes close to the apparent dip measurements.

23. Figure 6.13 shows a stereographic or equal angle projection of the apparent dip
data. This is very similar to the equal interval plot (Fig. 6.12) except that the
distance between the circles representing dip is not constant. Note, for example,
that the distance between the 0° dip and 10° dip circles is greater than the distance
between the 70° dip and 80° dip circles. In this case, the distance from the plot
centre is given by
r = R tan[(90 - Φ)/2]
Thus, a dip of 45° would plot a distance
r = R tan[(90 - 45)/2]
= R tan(22.5)
= 0.414 R
 which is significantly closer to the centre than half way out (i.e. closer to the
centre than in the equal interval plot). Applying Eqn. 6.2 to the cases of zero dip
and 90° dip gives a result of r = R and r = 0, respectively (the same as for the
equal interval plot).

24. Thank You!!!